## Geometric group theory seminar - 2014/15

### The seminar meets each Wednesday at 3 PM in BURN 920

#### October 1: Piotr Przytycki (McGill), *Overview of the article "Random groups contain surface subgroups"
of Calegari and Walker*

I will give an overview of random groups in the Gromow density
model, and the outline of the proof of a theorem of Calegari and Walker
that they contain surface subgroups.

#### October 8: Joseph Helfer (McGill), * The Thin Fatgraph Theorem*

Continuing the exposition of the paper "Random groups contain surface
subgroups" by Calegari and Walker, I will describe the Thin Fatgraph
Theorem and explain the part of its proof which reduces it to a
combinatorial problem.

#### October 15: Adam Wilks (McGill), *Annulus Moves and Pants Moves*

This talk will continue the discussion of Calegari's paper "Random Group contain Surface subgroups". In particular it will be show that a equidisributed collection of tagged loops of length 40L can be glued to a trivalent partial fatgaph having edge lengths at least L. The talk will be combinatorial in nature. This result is needed to finish the proof of the thin fatgraph theorem, introduced earlier in the paper.

#### October 22: Daniel Woodhouse (McGill), *Random One-Relator Groups Contain Surface Subgroups*

The Thin-Fatgraph Theorem will be applied to a one relator group. There will be a discussion of what a bead decomposition is, why it is needed, and how it can be guaranteed before concluding with a proof that the immersed surfaces we obtain will almost surely be

\pi_1-injective.

#### October 29: Dani Wise (McGill), *A survey on ordered groups*

Ordered groups have an amusing and sometimes critical role within combinatorial group theory that looks algebraic and is unaware of geometry but nevertheless often engages with simple topological ideas. Many interesting classes of groups are left-orderable. For instance this property holds for one-relator groups without torsion, braid groups, and knot groups. I will give a very perfunctory survey of this topic.

#### November 5: Dani Wise (McGill), *Ordering Trees*

I will describe the Dicks-Sunic ordering of the vertex set of a directed tree.
This gives an explicit left ordering of a free group, as well as the more general result that the free product of left orderble groups is left orderable.
Time permitting, I'll also describe the Magnus power series representation of a free group which gives an alternate explanation that is simpler but less elementary.

#### November 12: Nima Hoda (McGill), *Braid Groups are Left-Orderable*

I will present a proof that braid groups are left-orderable. A group
is left-orderable if it admits a total order that is invariant under
left multiplication (a < b implies ca < cb).

#### November 19: Micheal Brandenbursky (CRM), *Concordance group and stable commutator length in braid groups*

In this talk I will define quasi-homomorphisms from braid groups to the concordance group of knots and examine its properties and consequences of its existence. In particular, I will provide a relation between the stable four ball genus in the concordance group and the
stable commutator length in braid groups, and produce examples of infinite families of concordance classes of knots with uniformly bounded four ball genus. I will also provide applications to the geometry of the infinite braid group. In particular, I will show that its commutator subgroup admits a stably unbounded conjugation invariant norm. This answers an open problem posed by Burago, Ivanov and Polterovich. If time permits I will describe an interesting connection between the concordance group of knots and number theory. This work is partially joint with Jarek Kedra.

#### November 26, 2:30 PM: Marcin Kotowski (University of Toronto), *Random Schroedinger
operators with applications to Novikov-Shubin invariants*

For a finitely generated group G and a group ring element T, the
Novikov-Shubin invariant of T is a topological invariant related to
the spectral measure of T. We will consider lamplighter groups

Z_2
\wr Z and lattices in Sol group

Z^2 \rtimes Z, and show examples of
group ring elements with Novikov-Shubin invariants equal to zero. In
particular, this provides a simple finitely presented counterexample
to the Lott-Lueck conjecture about positivity of Novikov-Shubin
invariants. The main computational tool comes from the theory of
random Schroedinger operators. Joint work with Balint Virag.

#### November 26, 3:30 PM: Michal Kotowski (University of Toronto), *Non-Liouville groups with return probability exponent at most 1/2*

I will talk about constructing a finitely generated group

G without the Liouville property such that the return probability of a random walk satisfies

p_{2n}(e,e) \gtrsim e^{-n^{\gamma + o(1)}} for

\gamma = 1/2. This shows that the constant

1/2 in a recent theorem by Gournay, saying that return probability exponent less than

1/2 implies the Liouville property, cannot be improved. The construction is based on permutational wreath products over tree-like Schreier graphs and the analysis of large deviations of inverted orbits on such graphs. Joint work with Balint Virag.

#### December 3, 2014: Eduardo Martinez-Pedroza (Memorial), *A Subgroup Theorem for Homological Dehn Functions*

We use algebraic techniques to study homological Dehn functions of groups and their subgroups. These functions are (higher dimensional) homological versions of the standard isoperimetric function of a finitely presented group.
Our main result states that if

H< G are groups admitting finite

(n+1)-dimensional Eilenberg-Mclane spaces, then the nth-homological filling function of

H is bounded above by that of

G.
This contrasts with known examples where such inequality does not hold under weaker conditions on the group

G or the subgroup

H.
Joint work with Gaelan Hanlon.

#### January 28, 2015: Martino Lupini (York University), *An invitation to sofic groups*

The class of countable discrete groups known as sofic groups has drawn in
the last fifteen years the attention of an increasing number of
mathematicians in different areas of mathematics. Many long-standing
conjectures about countable discrete groups have been settled for sofic
groups. Despite the amount of research on this subject, several fundamental
questions remain open, such as: Is there any group which is not sofic? In
my talk I will give an overview of the theory of sofic groups and its
applications.

#### February 4, 2015: Joel Friedman (University of British Columbia), *Sheaves on Graphs, L^2 Betti Numbers, and Applications*

We will give an introduction to sheaves on graphs, and briefly discuss
their

L^2-Betti numbers and applications.
Our sheaves are sheaves of vector spaces on a topology naturally associated
to a graph. Roughly speaking, our setting can be regarded as a
generalization of algebraic graph theory, or as a natural, graph theoretic
example of a Grothendieck topology.
The merit of working with sheaves include (1) we can compare different
sheaves over the same graph via exact sequences, and (2) there are "new"
morphisms between sheaves representing graphs, i.e., morphisms between
graphs, viewed as sheaves, that do not occur as a morphism of graphs, and
(3) their "

L^2-Betti numbers" are integers and can be described in simple
terms.
This talk assumes only basic linear algebra and graph theory. Part of the
material is joint work with Alice Izsak and Lior Silberman.

#### February 11, 2015: Camille Horbez (Université de Rennes 1), *The Tits alternative for the automorphism group of a free product*

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*\dots*G_k*F_N,
where

F_N is a finitely generated free group, and all

G_i's are
nontrivial, non isomorphic to

\mathbb 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 a 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.

#### February 18, 2015: Pranab Sardar (UC Davis), *Packing subgroups in solvable groups.*

Bounded packing of subgroups of countably infinite groups were
defined by Chris Hruska and Dani Wise. A subgroup H of a group G has
bounded packing if any collection of pairwise uniformly close left cosets of H
is uniformly finite. This holds for separable subgroups of any group, quasi-
convex subgroups of hyperbolic groups and so on. Nevertheless, we only
know of a relatively few classes of groups and certain classes of subgroups
of them for which this is true. Hruska-Wise asks if this is true for all subgroups
of solvable groups.
In this talk, we will show that bounded packing holds for all subgroups of nilpotent-by-polycyclic groups. Finally, if time permits, we will describe an
example of a finitely generated solvable group of derived length 3 that has
a finitely generated subgroup without the bounded packing property.
Although the statements are geometric, proofs are mostly algebraic and
are easily accessible to all.

#### February 25, 2015: Kasia Jankiewicz (McGill University), *Incoherent Coxeter groups*

A group is coherent if every finitely generated subgroup has a finite presentation. Otherwise, it is incoherent. I will give a general overview of cohrerence. A powerful way of finding finitely generated subgroups with no finite presentation is to use Bestvina-Brady Morse theory. I will show how we used it to prove that certain Coxeter groups are incoherent. This is joint work with Dani Wise.

#### March 4, 2015, 11:30: Jingyin Huang (NYU), *Quasi-isometry classification of certain right-angled Artin groups
*

We show that two right-angled Artin groups G and G' are
quasi-isometric iff they are commensurable, given that G has finite outer
automorphism group. This is motivated by the previous rigidity result of
Bestvina-Kleiner-Sageev, and some of our ingredients can be used to
understand the asymptotic geometry of more general higher-rank CAT(0) cube
complexes. No previous knowledge about right-angled Artin group is required.

#### March 4, 2015, 3:00: Mark Hagen (University of Michigan), *Evolution of random right-angled Coxeter groups*

Given a natural number n, and a function p : N --> [0,1], a
"random graph of size n at density p" is produced by starting with n
vertices, and, for each pair of vertices, independently adding an edge
with probability p(n). It is often interesting to investigate, for a
property P of graphs, with what limiting probability a random graph of
size n and density p is P as n goes to infinity. An early result of
Erdos-Renyi established that, at density growing more slowly than
log(n)/n, a random graph is disconnected, and above that threshold,
connected. This is the first example of such a "sharp threshold
density" for some property to occur. Later, Klee-Larman and Bollobas
investigated the diameter of random graphs. We are in the process of
establishing sharp thresholds for two properties -- being an "augmented
suspension" (AS) and having a "component of full support" (CFS). The
former is a special case of the latter, and (AS) is to (CFS) as having
small diameter is to being connected (in a sense I'll make precise).
Interestingly, the threshold functions for these properties seem related
in a natural way to the thresholds for connectivity and small diameter;
I'll state our results, explain our suspicions, demonstrate a computer
program we've written to study random graphs, and talk about
applications to the geometry of Coxeter groups. This talk is on ongoing
joint work with Jason Behrstock and Tim Susse.

#### March 11, 2015: Stephane Lamy (Université Toulouse 3), *Small cancelation theory on non-locally compact spaces*

Recently small cancelation theory was extended to apply to
groups acting on non-locally compact hyperbolic spaces.
A first application is to the mapping class group G of a riemann
surface, acting on its curve complex: one can produce normal subgroups
in G that are free and purely pseudo-Anosov, with arbirary large stretch
factor (result by Guirardel, Dahmani, Osin).
A second application is to the Cremona group, the group of birational
transformations of the projective plane, which acts by isometry on a
infinite dimensional version of the usual hyperbolic disk (work by
Cantat and myself over an algebraically closed field, which was recently
extended to the case of any base field by my student Lonjou).
I will discuss the statement of the small cancelation criterion is this
setting and give an idea of the construction of the infinite dimensional
hyperbolic space in the second application.

#### March 18, 2015: Micheal Brandenbursky (CRM), *Quasi-morphisms in algebra and geometry I*

Quasi-morphisms on a group are real-valued functions which satisfy the homomorphism equation "up to a bounded error". They are known to be a helpful tool in the study of the algebraic structure of non-Abelian groups. In this lecture series, I will discuss constructions relating:

a) knots, braid groups, mapping class groups,

b) interesting metrics on groups of area-preserving diffeomorphisms of surfaces,

c) quasi-morphisms on groups of all such diffeomorphisms.

No previous knowledge of the subject will be assumed.

#### March 25, 2015, 2:45 PM: Micheal Brandenbursky (CRM), *Quasi-morphisms in algebra and geometry II*

Quasi-morphisms on a group are real-valued functions which satisfy the homomorphism equation "up to a bounded error". They are known to be a helpful tool in the study of the algebraic structure of non-Abelian groups. In this lecture series, I will discuss constructions relating:

a) knots, braid groups, mapping class groups,

b) interesting metrics on groups of area-preserving diffeomorphisms of surfaces,

c) quasi-morphisms on groups of all such diffeomorphisms.

No previous knowledge of the subject will be assumed.

#### March 25, 2015, 4:00 PM: Doron Puder (IAS), *Word Maps and Measure Preservation*

We establish new characterizations of primitive elements and free factors in free groups, which are based on the distributions they induce on finite groups.
More specifically, for every finite group G, a word w in the free group on k generators induces a word map from G^k to G. We say that w is measure preserving with respect to G if given uniform distribution on G^k, the image of this word map distributes uniformly on G. It is easy to see that primitive words (words which belong to some basis of the free group) are measure preserving w.r.t. all finite groups, and several authors have conjectured that the two properties are, in fact, equivalent. In a joint work with O. Parzanchevski, we prove this conjecture.

#### April 1, 2015: Jarek Kedra (University of Aberdeen), *The conjugation invariant geometry of cyclic subgroups*

Let G be a group generated by a set S which is a union of finitely many
conjugacy classes. Let |g| denote the word norm with respect to S. It is
conjugation invariant and its Lipschitz class does not depend on the
choice of a finite set. We are interested in the geometry of cyclic
subgoups, and more precisely in the growth rate of the sequence |g^n|,
where g is an element of G.
In the paper arXiv:1310.2921 we observed that for many classes of groups
the sequence |g^n| is either bounded or grows linearly. In other words,
the cyclic subgroup generated by g is either bounded or undistorted (this
is in contrast with classical geometric group theory where other types of
growth occur). Such a dichotomy holds for many classes of groups of
geometric origin (eg. braid groups, Coxeter groups, hyperbolic groups,
lattices in solvable Lie groups, lattices in some higher rank semisimple
groups, Baumslag-Solitar groups, right angled Artin groups etc).
Open problem: Find a finitely presented group which violates either
dichotomy.
Joint work with Misha Brandenbursky, Swiatoslaw Gal and Michal
Marcinkowski.

#### April 8, 2015: Micheal Brandenbursky (CRM), *Quasi-morphisms in algebra and geometry III*

Quasi-morphisms on a group are real-valued functions which satisfy the homomorphism equation "up to a bounded error". They are known to be a helpful tool in the study of the algebraic structure of non-Abelian groups. In this lecture series, I will discuss constructions relating:

a) knots, braid groups, mapping class groups,

b) interesting metrics on groups of area-preserving diffeomorphisms of surfaces,

c) quasi-morphisms on groups of all such diffeomorphisms.

No previous knowledge of the subject will be assumed.

#### April 15, 2015: Damian Orlef (University of Warsaw), *Random groups are not left-orderable*

I will show that nontrivial random goups obtained in the Gromov density model are not left-orderable at any density (with high probability). In the lecture I will present a technique of increasing the number of generators in the model, useful for proving properties which can be formulated in the setting of languages of finite automata.

Here is a link to our schedule in 2013/14