# Group theory seminar: 2013-2014 calendar

### Organizers: Mikael Pichot, Dani Wise

The seminar meets on Wednesdays at 3pm in BURN920.

### The Kervaire conjecture

Speaker: Dani Wise
Sept. 18, 2013, 3pm

### The Kervaire conjecture

Speaker: Dani Wise
Sept. 25, 2013, 3pm

### Homology in towers of covering spaces and solenoids

Speaker: Will Cavendish
Oct. 2, 2013, 3pm

### Homology in towers of covering spaces and solenoids

Speaker: Will Cavendish
Oct. 9, 2013, 3pm

### Modules controlled over CAT(0) spaces

Speaker: Ross Geoghegan (Binghamton)
Oct. 16, 2013, 3pm
Abstract: I will talk about recent joint work with Robert Bieri. Here, G is a group, M is a proper CAT(0) space on which G acts by isometries, and A is a finitely generated G-module which is controlled'' over M in a sensible way that will be explained. This situation leads us to a horospherical limit set Sigma(M,A) which is a subset of the boundary-at-infinity of M. This set, and some of its cousins, are of interest in themselves, and also because they are related to diverse parts mathematics: for example: (1) buildings (when M is a suitable symmetric space), (2) tropical varieties (when M is Euclidean n-Space), (3) geometrically finite groups of hyperbolic isometries (when M is Hyperbolic n-space). If time permits I may mention other interesting connections.

### Residual finiteness of outer automorphism groups

Speaker: Mathieu Carette
Oct. 23, 2013, 3pm

### Quasi-isometry vs commability for trees and free products

Speaker: Mathieu Carette
Oct. 30, 2013, 3pm
Abstract: What can be said about a finitely generated group quasi-isometric to a given metric space X? Answers to such questions may be called quasi-isometric rigidity results. A very strong answer could be that all finitely generated groups quasi-isometric to X are commensurable. However, commensurability is usually too restrictive. We shall discuss the weaker notion of commability, which accounts for much more quasi-isometric rigity results. In particular, we will determine its relationship with quasi-isometry in the cases of trees and free products. Our questions will lead us naturally to study the large scale geometry of locally compact groups.

### Bernoulli shifts and sofic groups

Speaker: Rob Graham
Nov. 6, 2013, 3pm

### The geometry of the bidisk

Speaker: Virginie Charette
Nov. 13, 2013, 3pm
Abstract: The bidisk is the product of two copies of the hyperbolic plane. While the bidisk has been often mentioned in the literature, notably as a basic example in the theory of symmetric spaces, its basic geometry has not been fully studied. The talk will be an elementary introduction to the bidisk, its isometries, as well as equidistant hypersurfaces in the bidisk. These arise when considering Dirichlet domains for isometric actions.

### TBA

Speaker: Daniel Woodhouse
Nov. 20, 2013, 3pm

### No seminar - US Thanksgiving

Nov. 27, 2013, 3pm

### The boundary action of an invariant random subgroup

Speaker: Jan Cannizzo
Dec. 4, 2013, 3pm
Abstract: Grigorchuk, Kaimanovich, and Nagnibeda recently investigated the ergodic properties of boundary actions of subgroups of the free group (drawing many parallels with the classical study of actions of Fuchsian groups on the boundary of the hyperbolic plane). We ask the question: What happens for invariant random subgroups? That is, what happens when our subgroup is chosen with respect to a conjugation-invariant probability measure on the lattice of subgroups of the free group?

We show that if the invariant random subgroup is sofic (roughly speaking, if it admits finite approximations), then its boundary action is conservative (there are no wandering sets). Central to our approach is the fact that much can be said in terms of the Schreier graphs associated with our subgroups. Time permitting, we may also discuss work in progress with Vadim Kaimanovich that bears on this question.

### (Non)relative hyperbolicity of random Coxeter groups

Speaker: Mark Hagen
Jan. 29, 2014, 3pm
Abstract: I'll first describe a result which characterizes relatively hyperbolic Coxeter groups, and shows that the peripheral subgroups enjoy a nice property called "thickness" (which I'll define). For right-angled Coxeter groups, this result becomes very concrete: one can tell whether or not the raCg associated to the finite graph G is relatively hyperbolic by examining the structure of G. I'll describe how to do this in polynomial time, and say some things about the prevalence of relative hyperbolicity for "random" right-angled Coxeter groups, i.e. those for which G is a random graph in the Erdos-Renyi model. This is joint work with Jason Behrstock, Pierre-Emmanuel Caprace, and Alessandro Sisto.

### Why should we care about braid groups?

Speaker: Michael Brandenbursky
Feb. 5th, 2014, 3pm
Abstract: In this talk I will give an introduction to braid groups and will show their relation to different branches of mathematics. No previous knowledge of the subject will be assumed.

### Computer driven theorems and questions in geometry

Speaker: Moira Chas (Stuny Brook)
Feb. 12, 2014, 3pm
Abstract: Given an orientable surface S with negative Euler characteristic, a minimal set of generators of the fundamental group of S, and a hyperbolic metric on S, each free homotopy class C of closed oriented curves on S, determines three numbers: the word length (that is, the minimal number of generators and inverses necessary to express C as a cyclically reduced word), the minimal self-intersection and the geometric length.

One the other hand, the set of free homotopy classes of closed directed curves on S (as a set) is the basis of a Lie algebra structure (discovered by Goldman). This Lie algebra is closely related to the intersection structure of curves on S.

These three numbers, as well was the Goldman Lie bracket of two classes, can be explicitly computed (or approximated) by the help of computer. These computations lead us find counterexamples to existing conjectures and to establish new conjectures.

For instance, we conjectured that the distribution of self-intersection of classes of closed directed curves on a surface with boundary, sampling by word length, appropriately normalized, tends to a Gaussian when the word length goes to infinity. Later on, jointly with Lalley, we proved this result. Recently, Wroten extended this result to closed surfaces.

In another direction, the computer allowed to us to study the relation between self-intersection of curves and length-equivalence. (Two classes a and b of curves are length equivalent if for every hyperbolic metric m on S, m(a)=m(b).)

Finally, the computer allowed us to probe conjectures about how the intersection structure of curves on S is "encoded" in the Goldman Lie algebra.

### Distorted Co-Dimension 1 Subgroups of Tubular Groups

Speaker: Daniel Woodhouse (McGill)
Feb 19, 2014, 3pm
Abstract: A tubular group is a graph of groups with cyclic edge groups and ZZ^2 vertex groups. In other words they are the fundamental groups of a graph of spaces with annular edge spaces and torus vertex spaces. In a recent paper of Wise it was shown how to obtain a wallspace if a tubular group acted freely on a CAT(0) cube complex. I will give a simple example of a wall that does not quasi-isometrically embed in a tubular space, and discuss how this fits in with other known results. There will be a discussion of what it means for a subgroup to quasi-isometrically embed for the neophytes.

### Poisson boundaries of CAT(0) cube complexes

Speaker: Michah Sageev (Technion)
Feb 26, 2014, 3pm
Abstract: TBA

### On Subgroups of Non-positively Curved Groups

Speaker: Eduardo Martinez Pedroza (Memorial)
March 12, 2014, 3pm
Abstract: In this talk I will discuss the following result: If C is a class of locally finite complexes closed under taking full subcomplexes and covers and G is the class of groups admitting proper and cocompact actions on one-connected complexes in C, then G is closed under taking finitely presented subgroups. As a sample application, the result implies that the following classes of groups are closed under taking finitely presented subgroups: groups acting geometrically on regular CAT(0) simplicial complexes of dimension 3, systolic groups, and groups acting geometrically on 2-dimensional negatively curved complexes. This is joint work with Richard G. Hanlon.

### Cubical small cancellation theory

Speaker: Kasia Jankiewicz (McGill)
March 19, 2014, 3pm
Abstract: Small cancellation conditions require the relations in a group presentation to have "small" overlaps with each other. The cubical small cancellation, a generalization of the classical theory, is due to Dani Wise and it was used by him in the proof of the Malnormal Special Quotient Theorem. In my talk, I will introduce cubical presentations and describe cubical small cancellation conditions, in parallel with the classical version. I will discuss the cubical version of Greendlinger's lemma, which is the fundamental theorem of this theory.

### On the one endedness of graphs of groups

Speaker: Nicholas Touikan (Carleton)
March 26, 2014, 3pm
Abstract: I will present a proof of a fact that everybody already knew was true: Let $A,B$ be groups with isomorphic subgroups $C_1 \leq A$ and $C_2 \leq B$ that are virtually cyclic, then an amalgamated free product $G=A*_{C_1=C_2}B$ is one ended if and only if $A, B$ are one-ended relative to $C_1, C_2$ respectively. The method that will be presented is actually quite general and can be used to analyse many ended graphs of groups. The proof is accessible to anybody who has an idea of the Bass-Serre tree for a free product with amalgamation or an HNN extension, an intuitive understanding of group ends, and who isn't afraid of square complexes.

### A-T-menability and Measured Wallspaces

Speaker: Daniel Woodhouse (McGill)
April 2, 2014, 3pm
Abstract: A-T-menable groups (or "Groups with the Haagerup property") have a continuous, isometric actions on Hilbert spaces which are metrically proper. The property is a "strong negation" of Kazhdan's Property (T). I will discuss one strategy for constructing metrically proper actions using wall spaces -- familiar to many from the theory of CAT(0) cube complexes. More generally the construction will work for measured wall spaces. As an example I will demonstrate that countable amenable groups are a-T-menable.