# Groups and Geometry in the South East

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

## 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.