# Groups and Geometry in the South East

Here are details of our previous meetings. Click here for the programme of our next meeting.

## Southampton, December 2 2022

The meeting will take place in building 2, room 1085, and you can find a map of the campus here.

#### 1:15-2:15: The conjugacy problem for ascending HNN-extensions of free groups

Alan Logan (St Andrews)

In this talk, I will explain how to solve the conjugacy problem for ascending HNN-extensions of free groups. In 2006, Bogopolski+Martino+Maslakov+Ventura solved the conjugacy problem for free-by-cyclic groups. Their proof is based on 2 key components, which are both proven using an analysis of free groups automorphisms via train-track maps. We follow this same route, but instead use an analysis of free group endomorphisms via the "automorphic expansions" of Mutanguha to prove the analogous 2 key components.

#### 2:30-3:30: Quasi-actions of groups on trees and quasi-trees

Jack Button (Cambridge)

A quasi-action of a group G on a metric space X associates a uniform quasi-isometry to each group element but these maps need not be bijective and so this need not be a genuine action, even if we just regard X as a set. We will review the definition and basic properties of quasi-actions with some examples. We then look at how we might turn a quasi-action into a genuine isometric action, which will require replacing X with some other space Y quasi-isometric to it on which G acts suitably by isometries. We conclude by taking X to be a (simplicial) tree and seeing what finiteness properties on G and on X are required to ensure that Y is also a tree.

#### 3:30-4:00: TEA

#### 4:00-5:00: Product set growth in mapping class groups

Alice Kerr (Bristol)

A standard question in group theory is to ask if we can categorise the subgroups of a group in terms of their growth. In this talk we will be asking this question for uniform product set growth, a property that is stronger than the more widely understood notion of uniform exponential growth. We will see how considering acylindrical actions on hyperbolic spaces can help us, and give a particular application to mapping class groups.

## UCL, October 28 2022

## Roberts Building G08 Sir David Davies LT, Roberts Building

### 1:15-2:15 Friedman-Ramanujan functions in random hyperbolic geometry

Laura Monk (Bristol)

The Weil-Petersson model is a very nice and natural way to sample random hyperbolic surfaces. Unfortunately, it is not easy to compute expectations and probabilities in this probabilistic setting. We are only able to compute expectations of quantities that depend on lengths of *simple* closed geodesics, i.e. geodesics with no self-intersections, thanks to breakthrough work by Mirzakhani. The aim of this talk is to present new ideas that allow to deal with non-simple geodesics, developed in an ongoing collaboration with Nalini Anantharaman. We show that certain averages can be expanded in powers of \(1/g\) and provide information on the terms appearing in this expansion. I will discuss the implications of these results in spectral geometry, and the inspiration we found in Friedman's work on random regular graphs.

### 2:30-3:30 Hyperbolic one-relator groups

Marco Linton (Oxford)

Since their introduction by Gromov in the 80s, a wealth of tools have been developed to study hyperbolic groups. Thus, when studying a class of groups, a characterisation of those that are hyperbolic can be very useful. In this talk we will turn to the class of one-relator groups. In previous work, we showed that a one-relator group not containing any Baumslag--Solitar subgroups is hyperbolic, provided it has a Magnus hierarchy in which no one-relator group with a so called `exceptional intersection' appears. I will define one-relator groups with exceptional intersection, discuss the aforementioned result and will then provide a characterisation of the hyperbolic one-relator groups with exceptional intersection. Finally, I will then discuss how this characterisation can be used to establish properties for all one-relator groups.

### 3:30-4:00 TEA in Malet Place Engineering Building 2.14

### 4:00-5:00 Dehn functions of coabelian subdirect products

Robert Kropholler (Warwick)

The class of coabelian subgroups of free groups provides many groups with interesting finiteness properties e.g. the Stallings-Bieri groups. For the finitely presented groups in this family we can study their Dehn functions. Various authors have worked on the case of the Stallings-Bieri groups. Building on these methods, particularly those of Carter and Forester, we give upper and lower bounds for Dehn functions of certain coabelian subgroups of direct products. This is joint work with Claudio Llosa Isenrich.

## Oxford, June 10 2022

Mathematical Institute, Room L5

### Property (T) and random quotients of hyperbolic groups

#### 1:30

Calum Ashcroft (Cambridge)

In his original manuscript on hyperbolic groups, Gromov asked whether random quotients of non-elementary hyperbolic groups have Property (T). This question was later refined by Ollivier, and then answered in the case of random quotients of free groups by Zuk (and Kotowski--Kotowski).

In this talk we answer the Gromov--Ollivier question in the affirmative. We will discuss random quotients and some of their properties, in particular with relation to Property (T).

### Connections between hyperbolic geometry and median geometry

#### 2:45

Cornelia Drutu (Oxford)

In this talk I shall explain how groups endowed with various forms of hyperbolic geometry, from lattices in rank one simple groups to acylindrically hyperbolic groups, present various degrees of compatibility with the median geometry. This is joint work with Indira Chatterji, and with John Mackay.

### TEA

#### 3:45

### Division, group rings, and negative curvature

#### 4:00

Grigori Avramidi (Bonn)

In 1997 Delzant observed that fundamental groups of hyperbolic manifolds with large injectivity radius have nicely behaved group rings. In particular, these rings have no zero divisors and only the trivial units. In this talk I will explain how to extend this observation to show such rings have a division algorithm (generalizing the division algorithm for group rings of free groups discovered by Cohn) and that these group rings have``freedom theorems’’ showing that all of their ideals that are generated by few elements are free, where the specific value of `few’ depends on the injectivity radius of the manifold (which can be viewed as generalizations from subgroups to ideals of some freedom theorems of Delzant and Gromov). This has geometric consequences to the homotopy classification of 2-complexes with surface fundamental groups and to complexity of cell structures on hyperbolic manifolds.

## Location: Warwick/Zoom March 18 2022

### Deciding when two curves are of the same type.

#### 1:30-2:20 (2.30-3.20 CET)

Thi Hanh Vo (Luxembourg)

Let \(S\) be a compact orientable connected surface with negative Euler characteristic. Two closed curves on \(S\) are of the same type if their corresponding free homotopy classes differ by a mapping class of \(S\). Given two closed curves on \(S\), we propose an algorithm to detect whether they are of the same type or not. This is joint work with Juan Souto.

### Commensurability of lattices in right-angled buildings

#### 2:30-3:20 (9.30-10.20 CST)

Sam Shepherd (Vanderbilt)

Given compact length spaces \(X_1\) and \(X_2\) with a common universal cover, it is natural to ask whether \(X_1\) and \(X_2\) have a common finite cover. In particular, are there properties of \(X_1\) and \(X_2\), or of their fundamental groups, that guarantee the existence of a common finite cover? We will discuss several examples, as well as my new result which concerns the case where the common universal cover is a right-angled building. Examples of right-angled buildings include products of trees and Davis complexes of right-angled Coxeter groups. My new result will be stated in terms of (weak) commensurability of lattices in the automorphism group of the building.

### The Steinberg representation

#### 3:30-4:20 (11.30-12.20 EST)

Andy Putman (Notre Dame)

The Steinberg representation is a topologically-arising representation that plays a basic role in geometry, arithmetic, topology, and group theory. A classical theorem of Steinberg and Curtis says that it is irreducible over finite fields. I will explain how to generalize this to infinite fields. This talk should be fairly elementary, and no background in representation theory will be assumed. This is joint work with Andrew Snowden.

## Location: Southampton/Zoom, December 10 2021

### On subdirect products of type \(FP_n\) of limit groups over Droms RAAGs

#### 1:30-2:30

Dessislava Kochloukova (University of Campinas, Brazil)

I am going to discuss one recent preprint (co-authored with J. Lopez de Gamiz Zearra) that was posted on arxiv in 2021. It is about limit groups over Droms RAAGs, subdirect products of such groups and their homological finiteness properties \(FP_n\).

### Helly groups and relative hyperbolicity

#### 2:45-3:45

Motiejus Valiunas (University of Wrocław)

A graph is said to be Helly if any collection of its pairwise intersecting balls has a non-trivial intersection, and we say a group is Helly if it acts geometrically on a Helly graph. Such an action implies various properties of a group, for instance, biautomaticity, contractibility of asymptotic cones, and the coarse Baum-Connes conjecture. Many well-known classes of groups, such as cocompactly cubulated and graphical C(4)-T(4) small cancellation groups, are Helly. Moreover, any hyperbolic group and any group acting cocompactly on a tree with Helly vertex stabilisers and finite edge stabilisers is known to be Helly; therefore, one may expect to find a relationship between relatively hyperbolic and Helly groups.

### Relatively Geometric Actions on CAT(0) Cube Complexes

#### 4:00-5:00

Eduard Einstein (University of Pittsburgh)

Daniel Groves and I introduced relatively geometric actions, a new kind of action of a relatively hyperbolic group on a CAT(0) cube complex. With mild assumptions on the peripheral groups, we showed that many of the favorable properties of proper cocompact actions of hyperbolic groups on CAT(0) cube complexes have natural relatively geometric analogues. For example, when the peripherals are residually finite, a group acting relatively geometrically is residually finite and full relatively quasiconvex subgroups are separable. In this talk, I will discuss background on relatively geometric actions, tools we use to construct relatively geometric actions and some of the favorable properties of groups that act relatively geometrically. As time permits, I will discuss an application to the Relative Cannon Conjecture and an application to small cancellation free products (joint work with Thomas Ng).

## Location: UCL/Zoom October 22, 2021

### Hyperbolic groups with non-hyperbolic subgroups of finite type

#### 1:30-2:30 UK TIME

Bruno Martelli (Pisa)

A natural question in geometric group theory is whether being hyperbolic is somehow a hereditary property of groups, that is one that is maintained on some (yet to be defined) well-behaved subgroups. We know from a result of Brady in 1999 that being finitely presented is not enough for a subgroup to inherit hyperbolicity, and we show here that being of finite type is also insufficient. We derive a counterexample quite directly from a construction of a hyperbolic 5-manifold that fibers over the circle. To build such a manifold we use some nice right-angled polytopes and Bestvina - Brady theory, heavily inspired from a paper of Janckiewicz, Norin, and Wise. This is joint work with Italiano and Migliorini.

### Hierarchical hyperbolicity of Artin groups of extra-large type

#### 2:45-3:45 UK TIME

Alexandre Martin (Heriot-Watt)

Artin groups form a class of groups generalising braid groups, and whose geometry remains elusive. In recent years, there has been a lot of progress in showing that certain classes of Artin groups are non-positively curved in an appropriate sense and display hyperbolic features. In this talk, I will explain how the action of Artin groups of extra-large type on their Deligne complex can be exploited to show that these groups are hierarchically hyperbolic. This talk is based on joint work with M. Hagen and A. Sisto.

### Stable Torsion Length

#### 4:00-5:00 UK TIME

Chloe Avery (Chicago)

The stable torsion length in a group is the stable word length with respect to the set of all torsion elements. We show that the stable torsion length vanishes in crystallographic groups. We then give a linear programming algorithm to compute a lower bound for stable torsion length in free products of groups. Moreover, we obtain an algorithm that exactly computes stable torsion length in free products of finite abelian groups. The nature of the algorithm shows that stable torsion length is rational in this case. As applications, we give the first exact computations of stable torsion length for nontrivial examples. This is joint work with Lvzhou Chen

## Location: Oxford/Zoom, June 18 2021

### Tits Alternative in dimension 2

#### 1:30-2:30PM

Piotr Przytycki (McGill)

A group G satisfies the Tits alternative if each of its finitely generated subgroups contains a non-abelian free group or is virtually solvable. I will sketch a proof of a theorem saying that if G acts geometrically on a simply connected nonpositively curved complex built of equilateral triangles, then it satisfies the Tits alternative. This is joint work with Damian Osajda.

### Coarse-median preserving automorphisms

#### 2:45-3:45

Elia Fioravanti (Bonn)

We study fixed subgroups of automorphisms of right-angled Artin and Coxeter groups. If Phi is an untwisted automorphism of a RAAG, or an arbitrary automorphism of a RACG, we prove that Fix(Phi) is finitely generated and undistorted. Up to replacing Phi with a power, the fixed subgroup is actually quasi-convex with respect to the standard word metric (which implies that it is separable and a virtual retract, by work of Haglund and Wise). Our techniques also apply to automorphisms of hyperbolic groups and to certain automorphisms of hierarchically hyperbolic groups. Based on arXiv:2101.04415.

### Some new CAT(0) free-by-cyclic groups

#### 4:00-5:00PM

Rylee Lyman (Rutgers-Newark)

I will construct several infinite families of polynomially-growing automorphisms of free groups whose mapping tori are CAT(0) free-by-cyclic groups. Such mapping tori are thick, and thus not relatively hyperbolic. These are the first families comprising infinitely many examples for each rank of the nonabelian free group; they contrast strongly with Gersten's example of a thick free-by-cyclic group which cannot be a subgroup of a CAT(0) group.

## Location: Warwick/Zoom, March 26, 2021

### Intersection of parabolic subgroups in large-type Artin groups

#### 1-2PM

Maria Cumplido (Sevilla)

Artin groups are a natural generalisation of braid groups from an algebraic point of view: in the same way that braids are obtained from the presentation of the symmetric group, other Coxeter groups give rise to more general Artin groups. There are very few results proven for every Artin group. To study them, specialists have focused on some special kind of subgroup, called "parabolic subgroups". These groups are used to build important simplicial complexes, as the Deligne complex or the recent complex of irreducible parabolic subgroups. The question "Is the intersection of parabolic subgroups a parabolic subgroup?" is a very basic question whose answer is only known for spherical Artin groups and RAAGs. In this talk, we will see how we can answer this question in Artin groups of large type, by using the geometric realisation of the poset of parabolic subgroups, that we have named "Artin complex". In particular, we will show that this complex in the large case has a property called sistolicity (a sort of weak CAT(0) property) that allows us to apply techniques from geometric group theory. This is a joint work with Alexandre Martin and Nicolas Vaskou.

### Limit dendrites for free group automorphisms.

#### 2:15-3:15PM

Jean Pierre Mutanguha (Max Planck)

The study of outer automorphisms of free groups borrows a lot of tools and ideas from the study of mapping classes of closed orientable surfaces. One tool that’s still missing is the canonical decomposition of mapping classes: up to isotopy, an orientation preserving surface homeomorphism preserves a unique minimal multicurve and the restriction to (orbits of) components of the multicurve’s complement is either a pseudo-Anosov or a finite-order homeomorphism. We will translate this canonical decomposition in terms of R-trees and then describe an analogue for exponentially growing outer automorphisms of free groups.

### The virtual section problem for the Mapping Class groups.

#### 3:30-4:30

Vladimir Markovic (Oxford)

The virtual section problem for the Mapping Class group \(\mathrm{Mod}(S)\) of a surface \(S\) asks whether there exists a section from G to \(\mathrm{Homeo}(S)\) (the group of homeomorphisms of \(S\)), where \(G<\mathrm{Mod}(S)\) is a finite index subgroup. I discuss the history of this problem including the most recent results by Chen-Markovic.

## Location: Southampton/Zoom, December 11, 2020

A zoom link will be sent to the list nearer the time.

### Incoherence of free-by-free and surface-by-free groups

#### 13:30-14:30 GMT

Genevieve Walsh (Tufts)

A group is coherent if every finitely generated subgroup is finitely presented, and incoherent otherwise. Many well-known groups are coherent: free groups, surface groups, and the fundamental groups of compact 3-manifolds. We consider groups of the form \(F_m \rtimes F_n\) or \(S_g \rtimes F_n\) where \(S_g\) is the fundamental group of a closed surface of genus \(g\). We show that all these groups are incoherent whenever \(g, n\) are at least 2, answering a question of D. Wise. One possible alternative method to prove incoherence would be to show that these groups virtually algebraically fiber. We additionally show that not all groups covered by our methods virtually algebraically fiber. This is joint work with Robert Kropholler and Stefano Vidussi.

### Crystallographic Helly Groups

#### 14:45-15:45 GMT

Nima Hoda (ENS Paris)

A Helly graph is a graph in which the metric balls form a Helly family: any pairwise intersecting collection of balls has nonempty total intersection. A Helly group is a group that acts properly and cocompactly on a Helly graph. Helly groups simultaneously generalize hyperbolic, cocompactly cubulated and C(4)-T(4) graphical small cancellation groups while maintaining nice properties, such as biautomaticity. I will show that if a crystallographic group is Helly then its point group preserves an \(L^{\infty}\) metric on \(\mathbb R^n\). Thus we will obtain some new nonexamples of Helly groups, including the 3-3-3 Coxeter group, which is a systolic group. This answers a question posed by Chepoi during the recent Simons Semester on Geometric and Analytic Group Theory in Warsaw.

### Asymptotics of Cheeger constants and unitarisability of groups

#### 16:00-17:00 GMT

Andreas Thom (Dresden)

Given a group Γ, we establish a connection between
the unitarisability of its uniformly bounded representations and the
asymptotic behaviour of the isoperimetric constants of Cayley graphs
of Γ for increasingly large generating sets. The connection hinges on
an analytic invariant Lit(Γ)∈[0,∞] which we call the Littlewood
exponent. Finiteness, amenability, unitarisability and the existence
of free subgroups are related respectively to the thresholds 0,1,2 and
∞ for Lit(Γ). Using graphical small cancellation theory, we prove that
there exist groups Γ for which 1

## Location: UCL/Zoom, October 23, 2020

A zoom link will be sent to the list nearer the time.

### Mapping class groups in genus at least 3 do not virtually surject to the integers

#### 1:00-2:00pm BST

Asaf Hadari (Hawaii)

Mapping class groups of surfaces of genus at least 3 are perfect, but their finite-index subgroups need not be - they can have non-trivial abelianizations. A well-known conjecture of Ivanov states that a finite-index subgroup of a mapping class group in genus at least 3 has finite abelianization. We will discuss a proof of this conjecture, which goes through an equivalent representation-theoretic form of the conjecture due to Putman and Wieland.

### JSJ decompositions and polytopes of hyperbolic one-relator groups

#### 2:15-3:15pm BST

Dawid Kielak (Oxford)

I will present a quadratic time algorithm computing the JSJ decomposition of a hyperbolic two-generator one-relator group with abelianisation of rank 2. The algorithm uses the Friedl--Tillmann polytope in a crucial way. (Joint work with Giles Gardam and Alan Logan.)

### Subgroups of right-angled Coxeter groups via Stallings-like techniques

#### 3:30-4:30pm BST

Pallavi Dani (Louisiana State)

Stallings folds have been extremely influential in the study of subgroups of free groups. I will describe joint work with Ivan Levcovitz, in which we develop an analogue for the setting of right-angled Coxeter groups, and use it to prove structural and algorithmic results about their subgroups.

## GGSE/NBGGT LMS Online Lectures: Fall 2020

In tandem with NBGGT (North British Geometric Group Theory Seminar) we have received funding from the London Mathematical Society to produce a series of geometric group theory lectures. These are given by given by early career researchers, aimed at beginning postgraduate students, and will take place over Zoom. If you wish to attend, please subscribe to the GGSE mailing list so you can receive the Zoom invitations when we send them out.

##### ALL GGSE LECTURES SHOULD TAKE PLACE AT 11AM. ALL NBGGT LECTURES SHOULD TAKE PLACE FROM 2-3PM. SUBSCRIBE TO THE MAILING LIST FOR THE MOST UP-TO-DATE INFORMATION.

### GGSE Lectures:

#### Property (T) etc: 5, 12 October

David Hume (Oxford)

#### Mapping Class Groups: 19, 26 October

Katie Vokes (IHES)

#### Separability and profinite techniques: 2, 9 November

Gareth Wilkes (Cambridge)

#### Hyperbolicity and Beyond: 16, 23 November

Benjamin Barrett (Bristol)

### NBGGT Lectures

#### Groups of automorphisms of rooted trees: Thursday 22nd and Thursday 29th October

Marialaura Noce (Gottingen)

#### Free groups via graphs: Thursday 5th and Thursday 12th November

Alan Logan (Heriot-Watt)

#### Growth in Groups: Thursday 19th and Thursday 26th November

Alex Evetts (Newcastle)

## Location: Oxford/Zoom, June 12

A zoom link will be sent to the list nearer the time

### Superrigidity in rank one and geometric applications

#### 2-2.45pm BST = 9-9.45am Eastern

Matthew Stover (Temple)

Margulis famously proved superrigidity of irreducible lattices in higher rank Lie groups, then used this to deduce that they are arithmetic. General superrigidity is known to fail for certain fundamental groups of hyperbolic manifolds, and there are nonarithmetic examples, but one still might wonder about the precise extent to which these fail. I will give a gentle introduction to what superrigidity means, then discuss work with Uri Bader, David Fisher, and Nicholas Miller on superrigidity of representations of fundamental groups of finite-volume real and complex hyperbolic manifolds that satisfy certain natural geometric conditions. Our main application is to prove arithmeticity finite-volume real and complex hyperbolic manifolds containing infinitely many "maximal" properly immersed totally geodesic submanifolds, thus, for example, the figure-8 knot complement is the only hyperbolic knot complement containing infinitely many properly immersed totally geodesic surfaces.

### Splitting Bianchi groups

#### 3.05-3.50pm BST = 9.05-9.50am Central

Alan Reid (Rice)

Let d be a square-free positive integer, and \(O_d\) the ring of integers in the imaginary quadratic number field \(Q(\sqrt{-d})\). The groups \(PSL(2,O_d)\) are known as the Bianchi groups, and have been long studied for the connections to number theory, geometry and topology. In this talk we consider the question of whether Bianchi groups can split over a co-compact Fuchsian subgroup, and prove that for large enough \(d\) they always do. Indeed we will sketch ideas in the proof that for large enough the Bianchi orbifold \(H^3/PSL(2,O_d)\) contains at least (constant)\(\cdot d\) closed orientable embedded totally geodesic 2-orbifolds. Some history, other results and applications will also be discussed time permitting. This is joint work with Junehyuk Jung.

### Filtered ends and obstructing group actions

#### 4.15-5pm BST = 9.15-10am Mountain

Emily Stark (Utah)

Studying the topology of a space at infinity offers a powerful perspective in geometric group theory. Filtered ends capture a space at infinity relative to a subspace: one considers complements of increasing neighborhoods of a subcomplex in a simplicial complex. The i-th homology groups of the complements form an inverse system, and the i-th Cech homology group is the associated inverse limit. The goal is then to compute the Cech homology groups of the filtered end and to use homological arguments to study these pairs of spaces. In this talk, I will explain how both goals are possible in the setting of coarse embeddings into coarse PD(n) spaces. Indeed, Kapovich--Kleiner proved a coarse Alexander Duality theorem to compute these deep homology groups. We extend their construction to prove a relative duality theorem, thus developing the homology theory further. As applications, one can use the deep homology groups together with homological arguments to prove that certain groups cannot act properly on a given manifold. This is joint work with Chris Hruska and Hung Cong Tran.

## Friday 28 February, 2020

Location: University of Warwick Mathematics Institute, Zeeman Building, Room B3.03.

### Please note that due to a mishap the timings have changed.

### 3:05-3:55 Homomorphisms between (and to) 3-manifold groups

Daniel Groves

After Perelman's proof of the Geometrization Conjecture, and the work of many others, the structure of compact 3-manifolds is by now quite well understood. Less well-understood are maps between 3-manifolds. Since many 3-manifolds are aspherical, maps between 3-manifolds (up to homotopy) can largely by understood via homomorphisms between their fundamental groups.

I will describe some general results which attempt to understand the structure of the set of homomorphisms from an arbitrary finitely generated group to the set of all 3-manifold groups. In particular, I will describe answers to questions of Agol and Liu and of Reid, Wang and Zhou.

### 4:00 Warwick Mathematics Colloquium B3.02

### 5:25-6:15 Dehn filling and the boundary of a relatively hyperbolic group

Jason Manning

I'll survey what is known about the way the boundary of a relatively hyperbolic group is affected by relatively hyperbolic Dehn filling. I'll talk both about geometric and algebraic topological properties of the boundary. Parts of this talk will be based on joint works with Groves, Groves-Sisto, and Wang.

### 6:20-7:10 (Non)-relative hyperbolicity of the separating curve graph

Katie Vokes

Given a compact, connected, orientable surface, we can define many associated graphs whose vertices represent curves or multicurves in the surface. A first example is the curve graph, which has a vertex for every simple closed curve in the surface and an edge joining two vertices if the corresponding curves are disjoint. We could alternatively restrict to those curves which separate the surface into two components. While the curve graph is known to always be Gromov hyperbolic, this is not the case for the separating curve graph. I will present joint work with Jacob Russell classifying for which surfaces the separating curve graph is hyperbolic, for which it is relatively hyperbolic, and for which it is neither of these.

## Friday 13 December, 2019

Location: Building 56 "Mathematics Student Centre", Highfield Campus, University of Southampton.

### 1:30-2:30 On Finitely Presented Groups that Contain Q

James Belk (University of St Andrews)

It is a consequence of Higman's embedding theorem that the additive group Q of rational numbers can be embedded into a finitely presented group. Though Higman's proof is constructive, the resulting group presentation would be very large and ungainly. In 1999, Pierre de la Harpe asked for an explicit and "natural" example of a finitely presented group that contains an embedded copy of Q. In this talk, we describe some solutions to de la Harpe's problem related to Thompson's groups F, T, and V. Moreover, we prove that there exists a group containing Q which is simple and has type F infinity. This is joint work with J. Hyde and F. Matucci.

### 2:40-3:40 Ramification structures for quotients of generalised Grigorchuk-Gupta-Sidki groups

Anitha Thillaisundaram (University of Lincoln)

Groups of surfaces isogenous to a higher product of curves can be characterised by a purely group-theoretic condition, which is the existence of a so-called ramification structure. Gul and Uria-Albizuri showed that quotients of the periodic Grigorchuk-Gupta-Sidki groups, GGS-groups for short, admit ramification structures. We extend their result by showing that quotients of generalisations of the GGS-groups also admit ramification structures, with some deviations for the case p=2. This is joint work with Elena Di Domenico and Sukran Gul.

### 3:40-4:00 TEA

### 4:00-5:00 Commensurability of Baumslag-Solitar groups

Alexander Zakharov (Chebyshev Laboratory, Saint Petersburg State University)

Two groups are called commensurable if they have isomorphic subgroups of finite index. In particular, finitely generated commensurable groups are quasi-isometric. Baumslag-Solitar groups form an interesting and important class of one-relator groups with unusual properties. While the quasi-isometry classification for them was known previously, due to Farb, Mosher and Whyte, the commensurability classification was not. In joint work with Montse Casals-Ruiz and Ilya Kazachkov we fill this gap by providing a complete commensurability classification of Baumslag-Solitar groups.

## Friday 25 October, 2019

Location: UCL, South Wing Garwood LT.

### 1:30-2:30: Homotopy type of the free factor complex

Radhika Gupta (Bristol)

The mapping class group of a surface acts on the curve complex which is known to be homotopy equivalent to a wedge of spheres. In this talk, I will define the 'free factor complex', an analog of the curve complex, on which the group of outer automorphisms of a free group acts by isometries. This complex has many similarities with the curve complex. I will present the result (joint with Benjamin Brück) that the free factor complex is also homotopy equivalent to a wedge of spheres. We will also look at higher connectivity results for the simplicial boundary of Outer space.

### 2:40-3:40: (Hierarchically) hyperbolic quotients of mapping class groups

Alessandro Sisto (Heriot-Watt)

The Dehn fillings of a relatively hyperbolic group are useful relatively hyperbolic quotients constructed in a certain way inspired by Thurston's hyperbolic Dehn filling theorem. In the context of Mapping class groups, a reasonable analogue of Dehn fillings are quotients by large powers of Dehn twists. I will discuss these and related quotients, which in particular provide many infinite hyperbolic quotients of mapping class groups in low complexity.

Based on joint works with Dahmani-Hagen and Hagen-Martin.

### 3:40-4:00: TEA

### 4:00-5:00: Action rigidity of free products of hyperbolic manifold groups

Daniel Woodhouse (Oxford)

Gromov's program for understanding finitely generated groups up to their large scale geometry considers three possible relations: quasi-isometry, abstract commensurability, and acting geometrically on the same proper geodesic metric space. A *common model geometry* for groups \(G\) and \(G'\) is a proper geodesic metric space on which \(G\) and \(G'\) act geometrically. A group \(G\) is *action rigid* if any group \(G'\) that has a common model geometry with \(G\) is abstractly commensurable to \(G\). We show that free products of closed hyperbolic manifold groups are action rigid. As a corollary, we obtain torsion-free, Gromov hyperbolic groups that are quasi-isometric, but do not even virtually act on the same proper geodesic metric space. This is joint work with Emily Stark.

## Friday 24 May, 2019

Locatation: Oxford, Mathematical Institute, Room L5.

### 1:15--2:15 Isoperimetric inequalities of Groups and Isoperimetric Profiles of surfaces

Panos Papasoglu

It is an interesting question whether Gromov's `gap theorem' between a sub-quadratic and a linear isoperimetric inequality can be generalized in higher dimensions. There is some evidence (and a conjecture) that this might be the case for CAT(0) groups. In this talk I will explain how the gap theorem relates to past work of Hersch and Young-Yau on Cheeger constants of surfaces and of Lipton-Tarjan on planar graphs. I will present some related problems in curvature-free geometry and will use these ideas to give an example of a surface with discontinuous isoperimetric profile answering a question of Nardulli-Pansu. (joint work with E. Swenson)

### 2:30--3:30 Conjugacy growth in groups, geometry and combinatorics

Laura Ciobanu

In this talk I will give an overview of what is known about conjugacy growth and the formal series associated with it in infinite discrete groups. I will highlight how the rationality (or rather lack thereof) of these series is connected to the geometry of groups such as (relatively) hyperbolic, groups acting on trees, or graph products, and how tools from analytic combinatorics can be employed in this context.

### 3:30--4:15 Tea/coffee

### 4:15--5:15 CAT(0) groups need not be biautomatic

Ian Leary

Ashot Minasyan and I construct (or should that be find?) examples of groups that establish the result in the title. These groups also fail to have Wise's property: they contain a pair of elements no powers of which generate either a free subgroup or a free abelian subgroup. I will discuss these groups.

## Friday 1 March, 2019

Location: University of Warwick, Zeeman Building. Talks 1 and 2 in MS.05, talk 3/colloquium in B3.02.

### 1:15-2:15 RAAGs and Stable Commutator Length

Nicholaus Heuer (Oxford)

Stable commutator length (scl) is a well established invariant of elements g in the commutator subgroup (write scl(g)) and has both geometric and algebraic meaning. Many classes of "non-positively curved" groups have a gap in stable commutator length: This is, for every non-trivial element g, scl(g)>C for some C>0. This gap may be thought of as an algebraic injectivity radius and can be found in many classes of 'negatively curved' groups, such as hyperbolic groups, Baumslag-Solitair groups, free products and Mapping Class Groups. However, the exact size of this gap usually unknown, which is due to a lack of a good source of quasimorphisms. In this talk I will construct a new source of quasimorphisms which yield optimal gaps and show that for Right-Angled Artin Groups and their subgroups the gap of stable commutator length is exactly 1/2.

### 2:30-3:30 Sofic approximations — what’s the problem?

Andreas Thom (Dresden)

I am planning to give a general introduction to sofic groups, mention a few applications to fundamental conjectures about groups and group rings, and explain Misha Gromov’s conjecture that all groups are sofic. Finally I want to present a natural generalization of Gromov’s conjecture due to Laszlo Lovasz and Balasz Szegedy that has recently been disproved in joint work with Gabor Kun.

### 4:00-5:00 GGSE/Warwick Mathematics Colloquium: Building more automatic structures for groups

Sarah Rees (Newcastle)

I'm talking about some new composition theorems for automatic groups, joint work with Hermiller, Holt and Susse, specifically relating to HNN extensions, amalgamated products, and more generally graphs of groups that are (coset) automatic relative to appropriate subgroups.

The concept of automaticity for a group was introduced by Thurston in the late 1980's, based on properties of the groups of compact hyperbolic 3-manifolds that had been identified by Cannon, which in particular facilitated computation with these groups. Automatic groups are finitely presented, with recognisable normal forms and word problem soluble in quadratic time, and if biautomatic they have soluble conjugacy problem. A number of properties of closure and composition for this class of groups were proved almost immediately after its definition; hence in particular the fundamental group of most (but certainly not all) compact 3-manifolds could be proved automatic.

I'll provide some background on the subject of automatic groups, providing some motivation and summarising what is known, what is open, and what cannot be true of groups in this class. I'll define Holt and Hurt's related concept of coset automaticity. And I'll provide some details of the methods we used to provide our recent results, and describe a few groups for which our results give automatic structures, where none were previously known.

### 5:00-6:30 Wine, _beer_, and cheese...

### 6:30-8:30 Dinner. Get in touch with Saul.

## Friday 30 November, 2018

Location: Ketley Room, Mathematical Sciences, Southampton Uni, 54 Level 4.

### 1:30-2:30 On the abelianization of (pure) big mapping class groups.

Javier Aramayona (Madrid)

A classical theorem of Powell asserts that the mapping class group of an orientable surface of finite topological type and genus at least three has trivial abelianization. The first part of the talk will be devoted to explaining a proof of this result, as well as discussing the remaining low-genus cases.

We will then show that, in stark contrast, mapping class groups of infinite-type surfaces can have infinite abelianization. More concretely, we will explain how to construct non-trivial integer-valued homomorphisms from mapping class groups of infinite-genus surfaces. Further, we will give a description the first integral cohomology group of pure mapping class groups in terms of the first homology of the underlying surface. This is joint work with Priyam Patel and Nick Vlamis.

### 2:40-3:40 Almost finitely presented subgroups of hyperbolic groups

Robert Kropholler (Tufts)

Hyperbolic groups form a well understood class of groups. However, the subgroups of hyperbolic groups can be much wilder. One might hope that by imposing extra conditions upon subgroups they become easier to understand. A positive result of Gersten shows that if G is hyperbolic and has cohomological dimension 2 and H is a subgroup of type FP_2, then H is hyperbolic. In particular, H is finitely presented.

I will detail work showing that this phenomenon is special to dimension 2 by constructing examples of subgroups of hyperbolic groups which are of type FP_2 but not finitely presented.

### 3:40-4:00 TEA

### 4:00-5:00 Ramanujan cubical complexes as higher-dimensional expanders.

Alina Vdovina (Newcastle)

Ramanujan graphs were first considered by Lubotzky, Phillips, Sarnak to get graphs with optimal spectral properties. In our days the theory of expander graphs and, in particular, Ramanujan graphs is well developed, but the questions is what is the best definition of a higher-dimensional expander is still wide open. There are several approaches, suggested by Gromov, Lubotzky, Alon and others, but the cubical complexes were not much investigated from this point of view. In this talk I will give new explicit examples of cubical Ramanujan complexes and discuss possible developments.

## Friday 26 October, 2018

Location: University College London, 25 Gordon Street, Room 706

### 1:30-2:30 Local maxima of the systole function

Maxime Fortier-Bourque (Glasgow)

The systole of a hyperbolic surface is the length of any of its shortest closed geodesics. Schmutz Schaller initiated the study of the systole function and its local maxima in the 90's. I will explain a construction of a new infinite family of closed hyperbolic surfaces which are local maxima for the systole. The simplest of these surfaces is the Bolza surface, which is the surface of genus 2 with the largest number of symmetries. In higher genus, we obtain super-exponentially many examples and most of them have a trivial automorphism group. This is joint work with Kasra Rafi.

### 2:40-3:40 Subgroups of hyperbolic groups

David Hume (Oxford)

Here there be dragons. But these dragons are not arbitrary, for instance, they have finite asymptotic dimension and therefore their Cayley graphs do not contain expanders. The purpose of this talk is to introduce new geometric invariants which quantify the statement "this Cayley graph does not contain expanders" and show that they provide new information on all finitely generated subgroups of hyperbolic groups. As an example we will use these invariants to provide a completely geometric proof that the only Baumslag-Solitar groups admitting a coarse embedding into a hyperbolic group are the virtually abelian ones.

### 3:40-4:00 Tea

### 4:00-5:00 Coarse geometry of finite quotients

Ana Khukhro (Cambridge)

Cayley graphs of finite quotients of a given group can capture a lot of information about the group if there are enough of them. Spaces created using these quotients display many interesting coarse-geometric properties and are often the first examples of spaces with a given property, since one can use group theory to control the geometry of these spaces. We will talk about the varied coarse geometry arising from quotients of free groups.

## Friday 1 June, 2018

Location: University of Warwick, Zeeman Building, B3.03

Times subject to change.

### 1:15-2:15 Integrals over unitary groups, maps on surfaces, and Euler characteristics

Michael Magee (Durham)

This is joint work with Doron Puder (Tel Aviv University). For a positive integer \(r\), fix a word w in the free group on r generators. Let \(G\) be any group. The word \(w\) gives a `word map' from \(G^r\) to \(G\): we simply replace the generators in w by the corresponding elements of \(G\). We again call this map \(w\). The push forward of Haar measure under \(w\) is called the \(w\)-measure on \(G\). We are interested in the case \(G = U(n)\), the compact Lie group of \(n\)-dimensional unitary matrices. A motivating question is: to what extent do the \(w\)-measures on \(U(n)\) determine algebraic properties of the word \(w\)?

For example, we have proved that one can detect the 'stable commutator length' of \(w\) from the \(w\)-measures on \(U(n)\). Our main tool is a formula for the Fourier coefficients of \(w\)-measures; the coefficients are rational functions of the dimension \(n\), for reasons coming from representation theory.

We can now explain all the Laurent coefficients of these rational functions in terms of Euler characteristics of certain mapping class groups. I'll explain all this in my talk, which should be broadly accessible and of general interest. Time permitting, I'll also invite the audience to consider some remaining open questions.

### 2:45-3:45 Sphere systems at the borders of outer space

Karen Vogtmann (Warwick)

Outer space is a contractible space on which the group \(\operatorname{Out}(F_n)\) of outer automorphisms of a free gorup acts properly but not cocompactly. Bestvina and Feighn defined a larger contractible space, called the bordification of Outer space, on which the action is both proper and cocompact. In joint work with K.-U. Bux and P. Smillie we found an equivariant deformation retract of Outer space homeomorphic to the bordification. In this talk I will interpret the boundary of this retract in terms of sphere systems in a doubled handlebody.

### 4:00-5:00 Colloquium

### 5:00-5:30 Wine and cheese

### 5:30-6:30 A pro-p curve complex and residual properties of the mapping class group

Gareth Wilkes (Oxford)

To study the finite quotient groups of the mapping class group it is natural to consider the outer automorphism groups of finite quotients of a surface group. Rather than study these individually, a better approach is to package the finite quotients together into a 'profinite group' which contains all the information of the finite groups in a potentially more tractable form. A more readily manipulated object is the 'pro-\(p\) completion', where one only considers finite groups with orders a prime power. In this talk, I will discuss the ways in which the pro-p completion of a surface group may 'split over a cyclic subgroup' in a certain sense, and the techniques by which such a classification is proved. This in turn allows the construction of a pro-p analogue of a curve complex, on which the outer automorphisms of the pro-p group act. From this action we may deduce a non-trivial residual property of the mapping class group.

## Friday 1 December, 2017

Location: Ketley Room, Mathematical Sciences, Southampton Uni, 54 Level 4.

### 12:00 Lunch

### 1:15 Horofunctions and groups of linear growth

Matthew Tointon (Neuchatel)

One can define a boundary of a metric space using certain functions called 'horofunctions'. When the metric space is a Cayley graph there is a natural action of the group on this boundary. Anders Karlsson has proposed a potential new method for proving Gromov's theorem on groups of polynomial growth using this action. In this talk I will show that such a method can be made to work in the case of groups of linear growth. Joint work with Ariel Yadin.

### 2:30 Conjugation invariant geometry of \(\operatorname{SL}(n,\mathbb{Z})\)

Jarek Kedra (Aberdeen)

It is known that the group \(\operatorname{SL}(n,\mathbb{Z})\) for \(n>2\) is boundedly generated by elementary matrices. It almost immediately follows that every conjugation invariant norm on \(\operatorname{SL}(n,\mathbb{Z})\) is bounded. In particular, the word norm associated with a conjugation invariant generating set has finite diameter. I will discuss the dependence of the diameter on the choice of a generating set and present applications to finite simple groups \(\operatorname{PSL}(n,q)\). This is a recent joint work with Assaf Libman and Ben Martin.

### 3:30 Tea

### 4:00 Exotic building lattices

Stefan Witzel (Bielefeld)

Lattices on buildings arise naturally as \(S\)-arithmetic groups. However, there are also lattices on exotic buildings, which cannot be \(S\)-arithmetic, and they will be the topic of the talk. Exotic lattices are interesting because they share certain properties with their arithmetic counterparts while they are intriguingly different in other respects. I will survey some of the recent results on exotic lattices and discuss some open questions

## Friday 27 **October**, 2017

Location: UCL Gordon House 106

### 1:30-2:30 Computing JSJ decompositions of hyperbolic groups

Ben Barrett (Cambridge)

When studying a group, it is natural and often useful to try to cut it up into simpler pieces. Sometimes this can be done in a canonical way analogous to the JSJ decomposition of a 3-manifold. When the group is hyperbolic, a canonical decomposition over virtually cyclic subgroups can be read from the topological features of the Gromov boundary of the group by a theorem of Bowditch. In my talk I will discuss the problem of detecting these topological features algorithmically, leading to the result that Bowditch's canonical JSJ decomposition of a hyperbolic group is computable.

### 2:40-3:40 Geodesic currents and counting curves

Viveka Erlandsson (Bristol)

A famous result by Mirzakhani gives the asymptotic growth of the number of curves on a hyperbolic surface of bounded length \(L\), as \(L\) grows. If \(S\) is a surface of genus \(g\) equipped with a hyperbolic structure, she showed that the number of such curves on \(S\) (in each mapping class group orbit) is asymptotic to a constant times \(L^{6g-6}\). In this talk I will explain, through the use of geodesic currents, why the same asymptotics hold for other notions of length. In particular, if one measures length with respect to any Riemannian metric on \(S\) or the word metric with respect to any finite generating set of its fundamental group.

### 3:40-4:00 TEA

### 4:00-5:00 Hyperbolicity in graphical small cancellation monsters

Dominik Gruber (ETH Zurich)

Infinitely presented small cancellation groups are a major source of examples and counterexamples in the theory of infinite groups. Gromov's graphical small cancellation theory is a generalization of classical small cancellation theory that has provided the only known groups with expander graphs embedded in their Cayley graphs. In my talk, I will discuss advances in understanding features of negative curvature of infinitely presented graphical small cancellation groups, such as admitting interesting actions on Gromov hyperbolic spaces. As an application, I will present a whole new small cancellation theory in varieties of \(n\)-periodic groups.

This is based on joint works with multiple co-authors.

## 19 May, 2017

Location: University of Oxford, Mathematical Institute

### 1:15 Commensurating actions of groups of birational transformations

Yves Cornulier (Orsay)

We introduce a natural action of the group of birational transformations of a variety, namely on the (suitably defined) set of hypersurfaces of all smooth projective models of this variety. In particular, for some groups with some restrictions on their commensurating actions (or equivalently on their actions on CAT(0) cube complexes) such as groups with Property FW, we obtain corresponding restrictions on their actions by birational transformations. This is joint work with Serge Cantat.

### 2:30 Degenerations of maximal representations, non-Archimedean upper half space and laminations

Alessandra Iozzi (ETHZ)

We study degenerations of maximal representations into \(Sp(2n,R)\) and identify phenomena already present in the Thurston boundary of Teichmüller apace as well as new geometric features. We give equivalent conditions for the existence of measured laminations in term of an appropriate notino of length. This is joint work with Marc Burger, Anne Parreau and Beatrice Pozzetti.

### 3:30 Tea

### 4:00 Recognition of fibring in compact 3-manifolds

Andrei Jaikin (Madrid)

Let \(M\) be a compact orientable 3-manifold. We show that if the profinite completion of \(\pi_1(M)\) is isomorphic to the profinite completion of a free-by-cyclic group or to the profinite completion of a surface-by-cyclic group, then \(M\) fibres over the circle with compact fibre.

## Friday 24 February, 2017

Location: University of Warwick, Zeeman Building, B3.03

### 1:15 Two shall be the number of the counting

Eric Swenson (BYU)

In ancient times, Brian Bowditch showed that if the boundary of a one-ended hyperbolic group has a local cut point \(p\), then then \(p\) is part of a cut pair, and the group virtually splits over virtually \(\Bbb Z\).

We show that if the boundary of a one ended CAT(0) group \(G\) is separated by a finite subset then it is separated by cut pair, and \(G\) virtually splits over virtually \(\Bbb Z\). We also examine nesting actions on \(\Bbb R\)-trees (because they wouldn't get out of the way).

### 2:45 Basmajian-type inequalities for maximal representations

Beatrice Pozzetti (Warwick)

An injective homomorphism of the fundamental group of an hyperbolic surface in the symplectic group \(Sp(2n,R)\) is a maximal representation if it maximizes the so-called Toledo invariant. Maximal representations form interesting and well studied components of the character variety generalizing the Teichmüller space, that is encompassed in the case \(n=1\). Basmajian's equality allows to compute the length of the boundary of a hyperbolic surface in term of the lengths of the orthogeodesics: geodesic segments orthogonal to the boundary at both endpoints. In joint work with Federica Fanoni we provide a generalization of this result to the setting of maximal representations.

### 4:00 Colloquium: Hausdorff Banach Tarski paradox and dense free subgroups

Emmanuel Breuillard (Muenster)

The smallest number of pieces required to duplicate a space \(X\) by transformations coming from a prescribed group \(G\) can be interpreted as a measure of how non-amenable the \(G\) action on \(X\) is. This number is 4 (the minimum possible) if and only if \(G\) contains a free subgroup acting on \(X\) with abelian stabilizers. Deligne and Sullivan (\(d\) odd) and Borel (\(d\) even), showed that this is the case when \(X\) is the \(d\)-dimensional sphere (\(d\) at least 2) and \(G\) the full group of its isometries. I will present a wide generalization of this fact, where \(G\) can be any finitely generated linear group acting on some algebraic variety \(X\). Joint work with B. Guralnick and M. Larsen.

## Friday 25 November, 2016

Location: Ketley Room, Mathematical Sciences, Southampton Uni, 54 Level 4.

### Lunch from 12 noon.

### 1:15PM Metabelian groups with large return probability

Lison Jacoboni (Orsay/Lyon)

Let \(G\) be a finitely generated group. The behaviour of the simple random walk on \(G\), associated to a finite generating set, is well-known for non-amenable groups or for groups of polynomial growth, thanks to works of Kesten and Varopoulos. Exponential growth groups only satisfy an upper bound in general. Many examples of groups reaching this bound are known - such as lamplighter group, polycyclic groups, solvable groups of finite rank - as of groups who do not. In this talk, I will recall what is known so far and I will describe a characterization for metabelian groups of this large behaviour. This can be formulated in terms of Krull dimension of certain submodules of the group.

### 2:30PM Stable categories of modules for infinite groups

Peter Symonds (Manchester)

Joint work with Nadia Mazza. We construct a stable module category for a large class of infinite groups using complete resolutions. This is similar to the construction of Tate cohomology. We then restrict to the group of endotrivial (or invertible) modules, in other words the Picard group. This has been intensely studied for finite groups. We develop enough machinery to allow us to make calculations for certain infinite groups.

### 3:30PM Tea

### 4:00PM Rational discrete first degree cohomology for totally disconnected locally compact groups

Ilaria Castellano (Southampton)

For a topological group \(G\) several cohomology theories have been introduced and studied in the past. In many cases the main motivation was to obtain an interpretation of the low-dimensional cohomology groups in analogy to discrete groups. The aim of this talk is to give firstly interpretations of the first degree rational discrete cohomology functor \(\mathrm{dH}^1(G,-)\) introduced in [3], where G is a totally disconnected locally compact (= t.d.l.c.) group. Secondly, it will be shown how these interpretations can be used to prove several results about t.d.l.c. groups in analogy with the discrete case. Namely, we prove that a non-trivial splitting of a compactly generated t.d.l.c. group can be detected by knowing a single cohomology group in analogy to [1, Theorem IV 6.10]. As a consequence, we characterize the compactly presented t.d.l.c. groups of rational discrete cohomological dimension 1 to be fundamental groups of a finite graph of profinite groups in analogy to [2, Theorem 1.1].

#### References

[1] Dicks, W. and Martin John Dunwoody. Groups acting on graphs. Vol. 17. Cambridge University Press, 1989.

[2] Dunwoody, M. J. Accessibility and groups of cohomological dimension one. Proceedings of the London Mathematical Society 3.2 (1979): 193-215.

[3] Castellano, I., and Th Weigel. Rational discrete cohomology for totally discon- nected locally compact groups. Journal of Algebra 453 (2016): 101-159.

## 28 October 2016

Location: Chadwick Lecture Theatre, B05 LT Chadwick Building, UCL

### 2:00PM Kähler groups and subdirect products of surface groups

Claudio Llosa Isenrich (Oxford)

A Kähler group is a group which can be realised as fundamental group of a compact Kähler manifold. I shall begin by explaining why such groups are not arbitrary and then address Delzant-Gromov's question of which subgroups of direct products of surface groups are Kähler. Work of Bridson, Howie, Miller and Short reduces this to the case of subgroups which are not of type \(\mathcal{F}_r\) for some \(r\). We will give a new construction producing Kähler groups with exotic finiteness properties by mapping products of closed Riemann surfaces onto an elliptic curve. We will then explain how this construction can be generalised to higher dimensions.

### 3:15 Exotic relatives of Thompson's group F

Yash Lodha (Lausanne)

Thompson's group F is a remarkable group discovered by Richard Thompson in the 1970s. It is like an atom, or a building block, for some families of groups that satisfy certain exotic group theoretic properties. In the study of these examples, there is an interplay between combinatorial group theory, topology, and dynamics. I will provide a survey of such groups that appear in my work (in part with some co-authors).

### 4:15 Tea

### 4:30 Coarse medians and Property A

Jan Spakula (Southampton)

Coarse medians were invented recently by Brian Bowditch (2011), with the aim of providing a common framework for both hyperbolic groups and mapping class groups. Loosely speaking, coarse median spaces are metric spaces which are locally approximable by CAT(0) cube complexes. The "rank one" situation corresponds exactly to the case of hyperbolic groups, which are (in the appropriate sense) locally approximable by trees (= rank one CAT(0) cube complexes).

Our main result is that coarse median spaces of finite rank have property A. This provides an alternative proof of the result of Y. Kida (2005) that mapping class groups have property A.

This is a joint work with Nick Wright.## 27 May 2016

Location: University of Oxford

Lectures will be held in Lecture Theatre 6 (L6)

### 2:00PM 1-relator quotients of free products

Jim Howie (Heriot-Watt)

One-relator groups are well-known to have many nice properties that are not shared by finitely presented groups on the whole.

If \(A:=\ast_{\lambda\in\Lambda}A_\lambda\) is a non-trivial free product of groups, and \(G\) is the quotient of \(A\) by the normal closure of a single element, it is natural to ask which, if any, of these nice properties are reflected in \(G\). For example, does each \(A_\lambda\) naturally embed in \(G\) (the Freiheitssatz)? If each \(A_\lambda\) has solvable word problem, is the same true for \(G\)?

These questions have been studied for many years. I will recall some of the older work, and report on some recent progress in joint work with Chinyere.

### 3:15PM Conjugacy separability of non-positively curved groups

Ashot Minasyan (Southampton)

Conjugacy separability, along with residual finiteness, are two classical residual properties of groups, which measure how well a given infinite group can be approximated by its finite quotients. These properties can be viewed as algebraic analogues of solvability of the conjugacy problem and the word problem in the group respectively.

For many groups residual finiteness can be shown quite easily (e.g., all finitely generated linear groups are residually finite). Conjugacy separability, on the other hand, is much harder to prove, and until recently only a few classes of conjugacy separable group were known.

During the talk I will discuss some approaches for establishing conjugacy separability of "non-positively curved" groups. I will present a proof that free groups are conjugacy separable and will outline an argument for showing that right angled Artin groups are conjugacy separable. I will then explain how the recent ground-breaking results of Haglund-Wise and Agol can be used to prove conjugacy separability of other large classes of groups (such as virtually compact special hyperbolic groups).

### 4:15PM Tea break

### 4:30PM Essential surfaces in graphs

Henry Wilton (Cambridge)

I'll describe some recent progress on the problem of finding surface subgroups in fundamental groups of graphs of free groups with cyclic edge groups.

## 26 February 2016

Location: University of Warwick

Lectures will be held in MS.04 (Zeeman Building)

### 1:15 A new cubulation theorem for hyperbolic groups

Daniel Groves (U. Illinois Chicago)

Agol proved that a hyperbolic group acting properly and cocompactly on a CAT(0) cube complex is virtually special. Wise's Quasiconvex Hierarchy Theorem says that a hyperbolic group acting cocompactly on a tree with quasiconvex and virtually special edge and vertex stabilizers is virtually special. I will talk about a common generalization of these two theorems: a hyperbolic group acting cocompactly on a CAT(0) cube complex with quasiconvex and virtually special cell stabilizers is virtually special. This is joint work with Jason Manning.

### 2:30 Ping Pong in CAT(0) cube complexes

Aditi Kar (Southampton)

I will describe some recent results of mine with Michah Sageev. Let \(G\) be a group acting properly and essentially on an irreducible, non-Euclidean finite dimensional CAT(0) cube complex X without a fixed point at infinity. We show that for any finite collection of simultaneously inessential subgroups \(\{H_1,\dotsc, H_k\}\) in \(G\), there exists an element \(g\) of infinite order such that, for all \(i\), the subgroup generated by \(H_i\) and \(g\) is the free product \(H_i*\langle g\rangle\). Examples of groups to which this applies are the Burger-Moses simple groups that arise as lattices in products of trees. We build a boundary of strongly separated ultrafilters and utilize the action of the group on the boundary to play ping pong. I will introduce all the terminology, describe the boundary if time permits and conclude with a summary of equivalent conditions for the reduced \(C^*\)-algebra of the group to be simple.

### 4:00 Break for departmental colloquim

Chéritat

### 5:00 Wine and cheese

### 5:30 Sphere boundaries of hyperbolic groups

Nir Lazarovich (ETH)

We show that the boundary of a one-ended simply connected at infinity hyperbolic group with enough codimension-1 surface subgroups is homeomorphic to a sphere. By works of Markovic and Kahn-Markovic our result gives a new characterization of groups which are virtually fundamental groups of hyperbolic 3-manifolds. Joint work with B. Beeker.

## 27 November 2015

Location: University of Southampton

### Pure acyclic complexes 1:15PM

Ioannis Emmanouil (Athens)

We shall review certain characterizations of the pure acyclic complexes of modules, which parallel the work of Neeman on the pure acyclic complexes of flat modules, and extend a result of Simson on modules that have a periodic pure resolution by pure projective modules.

### Quantifying residual properties of virtually special groups 2:30PM

Mark Hagen (Cambridge)

A subgroup \(H\) of a group \(G\) is said to be separable if, for each \(g\) in \(G\backslash H\), there is a finite-index subgroup \(G'\) of \(G\) that contains \(H\) but not \(g\). (If the trivial subgroup is separable, then \(g\) is "residually finite".) It is natural to ask what \([G:G']\) must be, in terms of the word-length of \(g\) and reasonable data about \(H\). This question amounts to estimating the "separability growth function" of \(G\), a special case of which is the "residual finiteness growth" introduced by Bou-Rabee. In this talk, I'll discuss upper bounds on these functions in the case where \(G\) is "virtually special" and \(H\) is quasiconvex. (For example, when \(G\) is the fundamental group of a surface and \(H=\{1\}\), we reprove the bounds on residual finiteness growth established independently by Patel and Rivin.) These results are joint with K. Bou-Rabee and P. Patel and rely on a high-dimensional version of Stallings's proof of Hall's theorem on subgroup separability of free groups.

### On characteristic modules of groups 4:00PM

Olympia Talelli (Athens)

We define a characteristic module for a group \(G\) to be a \(\mathbb{Z}G\)-module which is \(\mathbb{Z}\)-free , contains elements invariant under the action of \(G\) and has finite projective dimension over \(\mathbb{Z}G\). We relate the existence of such a module to the Gorenstein dimension of \(G\), the generalized cohomological dimension of G and proper actions of \(G\).

## Friday, October 23

Location: University College London, Chadwick G08

Getting to Chadwick G08: Find the main quad and pass through the doors at the southwest corner. Then turn left.

### 1:15 Expansion, Random Walks and Sieving in \(SL_2(\mathbb{F}_p\left[t\right])\)

Henry Bradford (Cambridge)

The past decade has seen remarkable developments in the construction of expander graphs from congruence images of linear groups over characteristic zero fields, beginning with the seminal work of Bourgain and Gamburd. These constructions have led in turn to many new results on return probabilities for random walks on such groups. We give the first application of Bourgain and Gamburd's methods to linear groups over fields of positive characteristic. We also highlight some potential obstructions to extending our results, which call for further investigation.

### 2:30 On a question of Bumagin and Wise

Alan Logan (Glasgow)

Bumagin and Wise asked if every countable group \(Q\) can be realised as the outer automorphism group of a finitely generated, residually finite group \(G_Q\). We further assume that \(Q\) is finitely generated. We give a partial solution when \(G_Q\) is recursively presentable, and a complete solution in this case modulo an open question of Osin. We give the first examples of groups \(G_Q\) where \(Q\) is not recursively presentable.

### 3:30-4:00 TEA

### 4:00 Anti-de Sitter 3-manifolds, Margulis spacetimes, and their higher-dimensional analogues

Fanny Kassel (Lille)

In 1983, Margulis constructed the first examples of proper affine actions of free groups on \(\mathbb{R}^3\). I will describe the geometry and topology of the corresponding quotients, which are flat Lorentzian 3-manifolds known as Margulis spacetimes, and of their negatively-curved counterparts, known as anti-de Sitter 3-manifolds. It is actually possible to perform similar constructions in higher dimension with a variety of discrete groups, not necessarily free. This is joint work with J. Danciger and F. Guéitaud.

## Friday, May 15, 2015

Location: University of Oxford

### 1:15PM Large scale geometry of Coxeter groups

Jason Behrstock (CUNY)

Divergence, thickness, and relative hyperbolicity are three geometric properties which determine aspects of the quasi-isometric geometry of a finitely generated group. We will discuss the basic properties of these notions and some of the relations between them. We will then then survey how these properties manifest in right-angled Coxeter groups and detail various ways to classify Coxeter groups using them.

This is joint work with Hagen and Sisto.

### 2:30PM The measurable Tarski circle squaring problem

Lukasz Grabowski (Warwick)

Two subsets A and B of R^n are equidecomposable if it is possible to partition A into pieces and rearrange them via isometries to form a partition of B. Motivated by what is nowadays known as Banach-Tarski paradox, Tarski asked if the unit square and the disc of unit area in R^2 are equidecomposable. 65 years later Laczkovich showed that they are, at least when the pieces are allowed to be non-measurable sets. I will talk about a joint work with A. Mathe and O. Pikhurko which implies in particular the existence of a measurable equidecomposition of circle and square in R^2.

### 4:00PM Splittings of free groups via systems of surfaces

Ric Wade (Utah)

There is a pleasing correspondence between splittings of a free group over finitely generated subgroups and systems of surfaces in a doubled handlebody. One can use this to describe a family of hyperbolic complexes on which Out(F_n) acts. This is joint work with Camille Horbez.

## Friday February 27, 2015

Location: University of Warwick

### 1:45-2:45 The Tits alternative for the automorphism group of a free product

Camille Horbez (Rennes 1)

A group G is said to satisfy the Tits alternative if every subgroup of G either contains a nonabelian free subgroup, or is virtually solvable. The talk will aim at presenting a version of this alternative for the automorphism group of a free product of groups. A classical theorem of Grushko states that every finitely generated group G splits as a free product of the form G_1*\dotsc*G_k*F_N, where F_N is a finitely generated free group, and all G_i's are nontrivial, non isomorphic to Z, and freely indecomposable. In this situation, I prove that if all groups G_i and Out(G_i) satisfy the Tits alternative, then so does the group Out(G) of outer automorphisms of G. I will present applications to proving the Tits alternative for outer automorphism groups of right-angled Artin groups, or of some classes of relatively hyperbolic groups. I will then present a proof of this theorem, in parallel to a new proof of the Tits alternative for mapping class groups of compact surfaces. The proof relies on the study of the actions of some subgroups of Out(G) on a version of the outer space for free products, and on a hyperbolic simplicial graph.

### 3:00-4:00 The structure of branch groups

Alejandra Garrido Angulo (Oxford)

Branch groups act faithfully on infinite rooted trees in a particular way, in some sense imitating the full automorphism group of the tree. The class of branch groups contains many examples with remarkable properties such as infinite finitely generated torsion groups, groups of intermediate word growth, amenable but not elementary amenable groups, etc. In the talk I shall explain how the subgroup structure of a branch group is determined by its action on a tree and how this structure in turn determines all possible branch actions. I will also show some applications.

### 4:00-5:30 Tea

### 5:30-6:30 Gradients in groups and applications

Nikolay Nikolov (Oxford)

Given a residually finite group G one can study the growth rate of various invariants attached to its subgroups of finite index e.g. homology with coefficients in Q or F_p, minimal number of generators, number of relations and so on. These growth rates have connections with diverse areas of mathematics: L^2 cohomology, measure preserving actions of G, profinite groups. In this talk I will survey the main results and central problems in this subject and discuss some recent progress.

## Friday 5th December, 2014

Location: Southampton

The Ketley Room, which is on Level 4 of the Mathematics Building: Building 54 of the Highfield Campus.

### 1:15PM Deviation estimates for random walks and acylindrically hyperbolic groups

Alessandro Sisto (ETH)

We will consider a class of groups that includes non-elementary (relatively) hyperbolic groups, mapping class groups, many cubulated groups and C'(1/6) small cancellation groups. Their common feature is to admit an acylindrical action on some Gromov-hyperbolic space and a collection of quasi-geodesics "compatible" with such action.

As it turns out, random walks (generated by measures with exponential tail) on such groups tend to stay close to geodesics in the Cayley graph. More precisely, the probability that a given point on a random path is further away than L from a geodesic connecting the endpoints of the path decays exponentially fast in L.

This kind of estimate has applications to the rate of escape of random walks (Lipschitz continuity in the measure) and its variance (linear upper bound in the length).

Joint work with Pierre Mathieu.

### 2:30PM Geometry of buildings

Petra Schwer (Karlsruhe)

During this talk I will show you some low-dimensional examples of spherical and affine buildings. Interesting geometric properties, such as retractions and decompositions, will be illustrated and we will see how they translate into group theoretic statements.

### 4:00PM Some aspects of the coarse geometry of the curve complex.

Vaibhav Gadre (Warwick)

Let S be an orientable closed surface. The curve complex associated to S is an infinite graph C(S) with vertices isotopy classes of simple closed curves with edges if the curves can be isotoped to be disjoint on S. The mapping class group has a natural action on C(S). Masur and Minsky showed that C(S) is Gromov-hyperbolic and the coarse geometry of C(S) has been of interest ever since. I will describe some aspects of it.

- pseudo-Anosov actions on C(S): Masur and Minsky showed that pseudo-Anosov maps act on C(S) by north-south dynamics. Bowditch showed that for a fixed genus the stable translation lengths form a discrete set. In joint work with Chia-yen Tsai we investigate the genus-dependence of the minimal stable translation length.
- Lipschitz constant: The systole map from Teichmuller space to C(S) associates to a marked hyperbolic surface the shortest curve on it. Masur-Minsky showed that the systole map is (K,C)-Lipschitz. In joint work with E.Hironaka, R. Kent and C. Leininger, we investigate the genus-dependence of the optimal Lipschitz constant.
- Train tracks: Masur and Minsky showed that the set of curves C(T) carried by a train track T on S is a quasi-convex subset of C(S). In joint work with S. Schleimer, we show that the complement C(S) - C(T) is also quasi-convex. This supports the intuition that when T is maximal C(T) is like a half-space in a Gromov hyperbolic space.

## Friday 17th October, 2014

Location: 25 Gordon Street, Room 706

### 1:15PM Classifying spaces for a family for hierarchically defined groups

Brita Nucinkis (Royal Holloway)

### 2:30PM The complex geometry of Teichmuller spaces and symmetric domains

Stergios Antonakoudis (Cambridge)

From a complex analytic perspective, Teichmuller space - the universal cover of the moduli space of Riemann surfaces - is a contractible bounded domain in a complex vector space. Likewise, Bounded Symmetric domains arise as the universal covers of locally symmetric spaces (of non-compact type). In this talk we will study isometric maps between these two important classes of bounded domains equipped with their intrinsic Kobayashi metric.

### 3:30PM-4:00PM Tea in 606 (Just downstairs.)

### 4:00PM Approximate groups and expanders

Ben Green (Oxford)

I will introduce the notion of an approximate group and explain some of the recent work that has been done on these. I will talk about how some of this work may be applied to show that all finite simple groups of Lie type may be generated by a pair {a,b} of elements extremely rapidly and uniformly: in fact, the random walk on {a,b, a^{-1}, b^{-1}} becomes equidistributed in time C log |G|, where C depends only on the Lie rank of the group G. Joint work with Emmanuel Breuillard, Bob Guralnick and Terry Tao.

## Friday 9th May, 2014

Location: Lecture Room 6, The Mathematical Institute, Andrew Wiles Building, Woodstock Road, Oxford.

### 1.15pm Cutting and pasting: a group for Frankenstein

Nicolas Monod (EPFL)

We know since almost a century that a ball can be decomposed into five pieces and these pieces rearranged so as to produce two balls of the same size as the original. This apparent paradox has led von Neumann to the notion of amenability which is now much studied in many areas of mathematics. However, the initial paradox has remained tied down to an elementary property of free groups of rotations for most of the 20th century. I will describe recent progress leading to new paradoxical groups.

### 2.30pm Embeddability between right-angled Artin groups and its relation to model theory and geometry

Montserrat Casals-Ruiz (Oxford)

In this talk we will discuss when one right-angled Artin group is a subgroup of another one and explain how this basic algebraic problem may provide answers to questions in geometric group theory and model theory such as classification of right-angled Artin groups up to quasi-isometries and universal equivalence.

### 4pm Some subgroups of topological Kac–Moody groups

Inna Capdeboscq (Warwick)

This talk is based on a joint work with B. Rémy (Lyon) in which we study some subgroups of topological Kac–Moody groups and the implications of this study on the subgroup structure of the ambient Kac–Moody group.

## Friday 21st February, 2014

There will be additional talks on the Thursday afternoon and Friday morning. Full details are available here. If you plan to attend, please register. Lunch will be provided on Friday—please indicate if you do not require lunch. There is also some funding available for accommodation; you can apply for this on the registration form.

Location: Warwick Mathematics Institute. The first two talks are in room MS.04, the final one in room B3.03.

### 1.15pm On the difficulty of inverting automorphisms of free groups

Enric Ventura (UPC)

We introduce a complexity function *α* (resp. *β*) to measure the maximal possible gap between the norm of an automorphism (resp. an outer automorphism) of a finitely generated group *G*, and the norm of its inverse. We shall concentrate in the case of free groups *F _{r}* and prove some results about the growth of these functions

*α*and

_{r}*β*: for rank

_{r}*r=2*,

*α*is quadratic and

_{2}*β*is linear; and for higher rank, we will give polynomial lower bounds for both functions, and a polynomial upper bound for

_{2}*β*(the lower bounds use just manipulation of automorphisms and counting techniques, while the proof of the upper bound makes use of a recent result by Algom-Kfir and Bestvina about the asymetry of the metric in the Outer Space). This is joint work with P. Silva and M. Ladra.

_{r}### 2.30pm Pro-*p* ends

Pavel Zalesskii (Brasilia)

We shall discuss a pro-*p* analogue of Stallings' theory of ends.

### 4pm Hyperbolic groups, Cannon–Thurston maps, and hydra

Tim Riley (Cornell)

Groups are Gromov-hyperbolic when all geodesic triangles in their Cayley graphs are close to being tripods. Despite being tree-like in this manner, they can harbour extreme wildness in their subgroups. I will describe examples stemming from a re-imagining of Hercules' battle with the hydra, where wildness is found in properties of "Cannon-Thurston maps" between boundaries. Also, I will give examples where this map between boundaries fails to be defined.

## Friday 6th December, 2013

Location: Building 67, room E1001, University of Southampton (maps and arrival information here)

### 1.15pm A polynomial upper bound on Reidemeister moves for each knot type

Marc Lackenby (Oxford)

For each knot type *K*, we establish the existence of a polynomial *p _{K}* with the following property. Any two diagrams of

*K*with

*n*and

*n'*crossings respectively differ by a sequence of at most

*p*(n) + p

_{K}_{K}(

*n'*) Reidemeister moves. As a consequence, the problem of deciding whether a knot is of type

*K*is in the complexity class NP. This result generalises earlier work which dealt with the case when

*K*is the unknot, for which we may take

*p*(

_{K}*n*) to be (231

*n*)

^{11}.

### 2.30pm Small cancellation groups and conformal dimension

John MacKay (Bristol)

The boundary at infinity of a hyperbolic group has a natural invariant
called its conformal dimension, introduced by Pansu. This analytic
invariant of the boundary can be studied using *l _{p}*-cohomology of the
group. I will discuss how recent work of Bourdon, Kleiner and others
combines with ideas of Ollivier and Wise to give new insights to the
geometry of small cancellation groups; in particular, to certain random
groups.

### 4pm Coarse embeddings of graphs and groups: monsters versus beauty

Goulnara Arzhantseva (Vienna)

The concept of coarse embedding was introduced by Gromov in 1993. It plays a crucial role in the study of large-scale geometry of groups and the Novikov higher signature conjecture. Coarse amenability, also known as Guoliang Yu's property A, is a weak amenability-type condition that is satisfied by many known metric spaces. It implies the existence of a coarse embedding into a Hilbert space. In this expository talk, we discuss the interplay between infinite expander graphs, coarse amenability, and coarse embeddings. We present several 'monster' constructions in the setting of metric spaces of bounded geometry.

This research was partially supported by my ERC grant ANALYTIC no. 259527.

## Friday 25th October, 2013

Location: Roberts 309, UCL

### 1.15pm Products of trees, quaternions and fake quadrics

Alina Vdovina (Newcastle)

We construct an infinite series of simply transitive irreducible lattices in PGL_{2}(**F**_{q}((t))) x PGL_{2}(**F**_{q}((t))) by means of a quaternion algebra over PGL_{2}(**F**_{q}((t))). The lattices depend on an odd prime power q = p^{r} and a parameter τ in **F**_{q}^{*} - 1, and are the fundamental groups of square complexes with just one vertex and universal covering T_{q+1} x T_{q+1}, a product of trees with constant valency q+1.
Our lattices give rise to smooth projective surfaces of general type over **F**_{q}. For q = 3, the Zariski–Euler characteristic attains its minimal value χ = 1: the surface is a non-classical fake quadric.

### 2.30pm A uniform bound for the bounded geodesic image theorem

Richard Webb (Warwick)

The bounded geodesic image theorem of Masur and Minsky states (in layman's terms) that efficient unmixing of complicated curves on a surface does not mix up too much on any proper subsurface. In the talk we shall concretely state the theorem and then give a proof with a universal bound. Some applications to mapping class groups will be discussed.

### 4pm The virtual fibering theorem for 3-manifolds

Stefan Friedl (Regensburg)

Ian Agol proved in 2007 that if an aspherical 3-manifold has virtually RFRS fundamental group, then the manifold is virtually fibered. We will present a variation of Agol's proof. This is joint work with Takahiro Kitayama.

## Friday 10th May, 2013

Location: Room RI.0.48, the Gibson Building, Oxford

### 1.15pm Homological dimension from an algebraic perspective

Peter Kropholler (Southampton)

A closer look at what we know about the homological (or weak) dimension of a group over various coefficient rings. This talk will survey the territory and include some discussion of the modern developments from ring theory and category theory.

### 2.30pm The solution to Siegel's problem on hyperbolic lattices

Gaven Martin (Massey University, Auckland, visiting Oxford)

We outline the history and the proof of the identification of the minimal covolume lattice of hyperbolic 3-space as the 3-5-3 Coxeter group extended by the involution preserving the symmetry of this diagram.

This solves (in three dimensions) the problem posed by Siegel in 1945.

Siegel solved this problem in two dimensions by deriving the signature formula identifying the 2-3-7 triangle group as having minimal co-area. There are strong connections with arithmetic hyperbolic geometry in the proof and the result has applications identifying three-dimensional analogues of Hurwitz's 84g-84 theorem as Siegel's result does.

### 4pm Ordering the space of finitely generated groups

Laurent Bartholdi (Göttingen)

Consider the following relation `emulates' between finite generated groups: *G* emulates *H* if, for some generating set *T* in *H* and some sequence of generating sets *S _{i}* in

*G*, the marked balls of radius

*i*in (

*G*,

*S*) and (

_{i}*H*,

*T*) coincide.

This means, informally, that any group-theoretical statement that can be computed in a finite portion of *H* can be computed in *G*.

Given a nilpotent group *G*, we characterize the groups that are related to *G* by the `emulation' relation: it consists, essentially, of those groups which generate the same variety of groups as *G*.

The `emulation' relation is transitive, so defines a preorder on the set of isomorphism classes of finitely generated groups. We show that a partial order can be imbedded in this preorder if and only if it is realizable by subsets of a countable set under inclusion.

We study the groups that emulate free groups. This lets us show that every countable group imbeds in a group of non-uniform exponential growth. In particular, there exist groups of non-uniform exponential growth that are not residually of subexponential growth and do not admit a uniform imbedding into Hilbert space.

This is joint work with Anna Erschler.

## Friday 22nd February, 2013

Location: Chadwick Lecture Theatre, B05 Chadwick Building, UCL

### 2.30pm Subset currents on free groups

Tatiana Nagnibeda (Geneva)

I'll talk about a new notion of a "subset current" on a hyperbolic group—a natural generalization of geodesics currents studied by Bonahon and others. We'll discuss in more detail the structure of the space of subset currents in the case of a free group. Joint work with Ilya Kapovich.

### 4pm Infinitesimal structure of simple locally compact groups

Pierre-Emmanuel Caprace (Louvain)

This talk concerns compactly generated simple locally compact groups. The case of connected groups is well understood: it corresponds exactly to the simple Lie groupes. The complementary case is that of totally disconnected groups. The goal of the talk is to illustrate that, if one excludes discrete groups, the groups in question carry a surprisingly rich structure, notably from a dynamical viewpoint, which arises from a study of arbitrary small identity neighbourhoods. This is based on a joint work with Colin Reid and George Willis.

## Friday 7th December, 2012

Location: Building 67, room E1001, University of Southampton (maps and arrival information here)

### 1.30pm Generalised triangle groups

James Howie (Heriot–Watt)

A generalised triangle group is one given by a presentation of the
form < x,y | x^{p} = y^{q} = W(x,y)^{r} = 1 > for some word W and integers p,q,r > 1. A long-standing conjecture of Rosenberger says that such groups satisfy a form of Tits alternative - either the group is virtually soluble, or it contains a rank-2 free subgroup. I will give a survey of this problem and sketch some recent results.

### 2.30pm Outer billiards

Richard Schwartz (Brown)

I'll survey what is known about outer billiards, especially the polygonal case. In particular, I'll give some computer demos of my solution to (what had been) a main question in the subject, known as the Moser–Neumann problem, which asked about the existence of unbounded orbits.

### 4pm Quasi-isometries of groups admitting certain cyclic JSJ decompositions

Christopher Cashen (Vienna)

A common problem in geometric group theory has been to show that various different kinds of decompositions of groups into pieces are respected by quasi-isometries. I will talk about the converse problem of deciding when groups constructed by gluing together similar pieces are quasi-isometric.

## Friday 19th October, 2012

Location: Chadwick Lecture Theatre, B05 Chadwick Building, UCL

### 2.30pm Analytic properties of small cancellation groups

Cornelia Druţu (Oxford)

Infinitely presented groups, especially those constructed using small cancellation techniques, are a rich source of counterexamples, in particular for properties implying the Baum–Connes Conjecture.
This talk (on joint work with Goulnara Arzhantseva) is about positive results on infinitely presented small cancellation groups: such groups satisfy the property of Rapid Decay (relevant to the Baum–Connes Conjecture), the Grothendieck metric approximation property, they are weakly amenable and C^{*}-exact. This provides the first examples of infinitely presented groups with the property of Rapid Decay (with the metric approximation property) among direct limits of Gromov-hyperbolic groups.

### 4pm Homotopy equivalences between nilpotent varieties

Juan Souto (British Columbia)

We study the relation between the representation spaces Hom(Γ,G) and Hom(Γ,K) where Γ is a finitely generated nilpotent group, G is an algebraic group and K is a maximal compact subgroup of G. For G "reductive" and Γ "expanding" we prove that these varieties are homotopy equivalent. On the other hand, we observe that at least one of the two adjective in quotation marks is needed. This is joint work with Lior Silberman.

## Friday 4th May, 2012

Location: Room 130, Huxley Building, Imperial College London.

### 2.45pm Realisation and dismantlability

Piotr Przytycki (Warsaw)

This is joint work with S. Hensel and D. Osajda. We give a new proof of the Nielsen Realisation Problem for a punctured surface: any finite subgroup of the mapping class group of a punctured surface acts as isometries of some hyperbolic metric. Our method is to find a fixed clique of the action of the finite group on the arc graph, using its "dismantlability". This approach also shows that the set of fixed points in Harer's spine is contractible. The strategy works for actions on the disc and sphere graphs as well.

### 4pm Cocompact lattices on Ã_{2} buildings

Anne Thomas (Sydney)

An Ã_{2} building is a CAT(0) polygonal complex which is a union of Euclidean planes tessellated by equilateral triangles. If K is the field of formal Laurent series over a finite field, and G = SL(3,K), then there is an Ã_{2} building X on which G acts with quotient a triangle and compact stabilisers. A cocompact lattice in G is then a group which acts on X cocompactly with finite stabilisers. We construct new cocompact lattices in G and relate them to previous examples. Our methods include extending work of Cartwright, Mantero, Steger and Zappa, which used cyclic simple algebras, and considering the action of finite groups of Lie type on X. This is joint work with Inna Capdeboscq and Dmitri Rumynin.

## Friday 17th February, 2012

Location: The Gibson Building, Oxford.

### 1.40pm Automorphisms of free groups, hairy graphs and modular forms

Karen Vogtmann (Cornell)

Abstract: The group of outer automorphisms of a free group acts on a space of finite graphs known as Outer space, and a classical theorem of Hurwicz implies that the homology of the quotient by this action is an invariant of the group. A more recent theorem of Kontsevich relates the homology of this quotient to the Lie algebra cohomology of a certain infinite-dimensional symplectic Lie algebra. Using this connection, S. Morita discovered a series of new homology classes for Out(F_{n}). In joint work with J. Conant and M. Kassabov, we reinterpret Morita's classes in terms of hairy graphs, and show how this graphical picture then leads to the construction of large numbers of new classes, including some based on classical modular forms for SL(2,Z).

### 2.30pm Cocompact actions for arithmetic groups

Stefan Witzel (Münster)

Abstract: We will survey constructions of cocompact actions for (S-)arithmetic groups and discuss how these can be used to prove finiteness properties.

### 4pm Central extensions, bounded cohomology and stable commutator length

Indira Chatterji (Orléans)

Abstract: I will explain basic facts about central extensions and subgroup distortion, and in particular the well-known fact that a central extension given by a bounded 2-cocycle gives an undistorted center. The converse is an open problem in general but is completely understood in the case of connected Lie groups. We show that the fundamental group of a connected Lie group is undistorted in its universal cover if and only if each integral Borel cohomology class of a connected Lie group G can be represented by a Borel bounded cocycle. We also investigate the cases where all elements of the fundamental group are undistorted (but not necessarily the whole group) and give an equivalent condition in terms of stable commutator length. This is joint work with Y. Cornulier, G. Mislin and C. Pittet.

## Friday 2nd December, 2011

Location: H O Schild Pharmacology Lecture Theatre, UCL.

### 2.30pm The cohomological dimension of the hyperelliptic Torelli group

Tara Brendle (Glasgow)

Abstract: The hyperelliptic Torelli group SI(S) is the subgroup of the mapping class group of a surface S consisting of elements which commute with a fixed hyperelliptic involution and which act trivially on homology. The group SI(S) appears in a variety of settings, including in the context of the period mapping on the Torelli space of a Riemann surface and as a kernel of the classical Burau representation of the braid group. We will show that the cohomological dimension of SI(S) is g-1; this result fits nicely into a pattern with other subgroups of the mapping class group, particularly those of the Johnson filtration. This is joint work with Childers and Margalit.

### 4pm The Simple Loop Conjecture for limit groups

Lars Louder (Oxford/Michigan)

Abstract: There are noninjective maps from surface groups to limit groups that don't kill any simple closed curves. As a corollary, there are noninjective all-loxodromic representations of surface groups in SL(2,C) that don't kill any simple closed curves, answering a question of Minsky. There are also examples, for any k, of noninjective all-loxodromic representations of surface groups killing no curves with self intersection number at most k.

## Friday 28th October, 2011

Location: H O Schild Pharmacology Lecture Theatre, UCL.

### 2.30pm On the growth of Betti numbers of locally symmetric spaces

Nicolas Bergeron (Jussieu)

Abstract : We study homology growth of lattices in a simple Lie group G. One is lead to a notion of local convergence of lattices or more generally of invariant random subgroups of G. After recalling some basic notions of locally symmetric spaces and lattices in Lie groups I will define local convergence and explain its relationship with homology growth. The situation is very nice when G has R-rank at least 2: any infinite sequence of lattices converges locally to the identity subgroup! When G has rank one this is no longer true but at least one has local convergence of congruence subgroups inside a given cocompact arithmetic subgroup of G. This is part of joint work with Abert, Biringer, Gelander, Nikolov, Raimbault and Samet.

### 4pm Farrell–Hsiang groups

Arthur Bartels (Münster)

Abstract: Farrell–Hsiang used a beautiful combination of controlled topology and Frobenius Induction to prove that the Whitehead group for fundamental groups of flat Riemannian manifolds is trivial. In this talk I will revisit this method and discuss a wider context where it can be used. Ultimately this plays a role in the computation of K- and L-theory of group rings over cocompact lattices in virtually connected Lie groups.