Montreal Geometric & Combinatorial Group Theory Seminar

The seminar meets each Wednesday at 3:30pm in 920 Burnside Hall at 805 Sherbrooke West - McGill University.

">(Here is a link to our schedule during 2007-2008)

(Here is a link to our schedule during 2005-2006)

(Here is a link to our schedule during 2004-2005)

Fall 2006:

September 20 – Dani Wise (McGill)

Cubulating Arithmetic Groups”

We describe the following result and its consequences:
Theorem: Let G be an arithmetic hyperbolic group of simple type.
Then G acts properly on a locally-finite CAT(0) cube complex.
This is joint work with Frederic Haglund.

September 27 – Olga Kharlampovich (McGill)
“Groups acting on trees”
In this talk I will discuss some methods and techniques designed
to deal with groups acting freely on $\Lambda$-trees. These methods
were extensively, though sometimes implicitly,  used in our (joint
with Alexei Myasnikov) solution of the Tarski's problems. It seems
it is worthwhile  to introduce them explicitly. Our key players in
this area are  infinite non-Archimedean words and Elimination
Processes. I will discuss joint results with A. Myasnikov and D. Serbin.

October 4 – No Seminar

October 11 – Genevieve Walsh (UQAM & Tufts)

“Commensurability classes of 2-bridge knots”

Two 3-manifolds are said to be commensurable if they have a common

finite-sheeted cover. Commensurability classes are a reasonable way to

organize hyperbolic 3-manifolds. For example, if a manifold is

virtually fibered or virtually Haken, then so is every manifold in its

commensurability class. However, the general problem of determining if

two hyperbolic 3-manifolds are commensurable is difficult. We show

that a hyperbolic 2-bridge knot complement is the unique knot

complement (in S^3) in its commensurability class. The proof relies

heavily on facts particular to 2-bridge knots.

There are commensurability classes that contain more than one

hyperbolic knot complement. For example, this can happen if one of the

knots admits a lens space surgery. We speculate on the general case.

This is joint work with Alan Reid.

October 18 – Nicholas Touikan (McGill)

“A fast Algorithm for Stalling's Folding Process”

Stalling's folding process is  a key ingredient in the solution

of many algorithmic problems involving subgroups of a free group. After

surveying some applications I will present an algorithm which

(theoretically) performs this process in worst case time O(N log^*(N)).

October 25 – David Janzen (McGill)

“A Story of Four Squares and an Anti-Torus

In this talk, we provide an example of a complete square complex built

from only four squares whose universal cover contains a particular type of

aperiodic flat called an anti-torus.  Such anti-tori have been used

previously by Wise to answer many questions concerning, for example, the

residual finiteness of fundamental groups of compact non-positively curved

2-complexes and the nature of flats appearing in the universal covers of

such complexes.  Complete square complexes provide many simple examples of

2-dimensional non-positively curved spaces.  Many questions about compact

non-positively curved spaces prove difficult to answer even when they are

restricted to questions about complete square complexes.

November 1 – Denis Serbin (McGill)

F.p. groups with regular free length functions.”

We introduce an analog of Makanin-Razborov

process in free groups for f.p. groups with regular

free length functions in an arbitrary ordered abelian

group. This process rewrites the set of relators and

reveals the structure a group. In particular, we show

that f.p. groups with regular free length functions

are Z^n-free.

November 8 – Ilya Kazachkov (McGill)

“On the automorphism group of right-angled Artin groups”

To a graph G is associated a partially commutative group P

(alias right-angled Artin group). First we develop an orthogonality

theory for graphs. This is a key tool for describing of the

centraliser of an arbitrary subset of P and the centraliser lattice of

P. As applications we obtain a description of the structure of the

automorphism group of P.

November 15 – John Labute (McGill)

"Tame pro-p-groups"

We introduce a new family of finitely presented pro-p-groups

which we call tame. These groups have cohomological dimension 2, have

exponential growth and subgroups of finite index have finite

abelianizations. While they occur remarkably often as Galois groups of

maximal p-extensions of number fields unramified outside a finite set of

primes with residual characterisics different from p, there is no

example of such a group whose presentation is known.

December 6 – Nicholas Bergeron (Paris - Sud)

“Arithmetic hyperbolic manifolds:

how to construct them and how they (should) look”.

Abstract : I will first explain how to construct hyperbolic manifolds in any

dimension using arithmetic groups. Above dimension 5, these are essentially

the only negatively curved manifolds known (up to surgery and trivial

modification of the metric). One may ask what kind of topology one thus gets.

In the second part of my talk I'll briefly review known and

conjectural properties of the (co)homology groups of these arithmetic

hyperbolic manifolds.

Winter 2007:

Jan 10 - Alexei Miasnikov  (McGill) "Zero-one laws and random subgraphs of  Cayley graphs"
I am going to discuss asymptotic properties of finite subgraphs of a fixed Cayley graph Gamma.
It turns out that the classical Zero-One law also holds in this situation, i.e., for any first-orfer
sentence either this sentence or its negation holds with probability one on finite subgraphs of Gamma.
One of the key ingredients of our approach is to show that the random subgraphs of  Gamma
(those that have a non-zero probability to occur) are  all elementarily equivalent to each other.
This brings  some interesting connections with the theory of percolation on groups.
(joint With R.Gilman and Yu.Gurevich)
The talk is elementary and self-contained.

Tuesday Jan 16, Chris Hruska (Milwaukee) “Relative hyperbolicity of countable groups”.

In the 1980s, Gromov promoted the idea of studying finitely

generated groups as metric spaces, using the word metric for

a finite generating set.  In fact, arbitrary countable groups are also

natural geometric objects.  Each countable group admits a proper,

left invariant metric.  This elementary idea can be used to extend many

coarse'' geometric techniques from finitely generated groups

to countable groups.

As an application, we prove that various notions of relative hyperbolicity

are equivalent for countable groups.  This equivalence was previously

understood only in the finitely generated case (by work of Bowditch,

Osin, and others). I will also discuss a substantial clarification of the notion of

a quasiconvex subgroup of a relatively hyperbolic group.

These are the most geometrically natural subgroups, and are themselves

relatively hyperbolic. Yet until now their basic study has been hindered by the fact that they

are often not finitely generated.

Jan 24 - Vyacheslav Futorny (Sao Paolo) "Noncommutative orders in skew group rings"

We will discuss the construction and properties of Galois subalgebras in skew

(semi)group rings. These algebras can be viewed as a noncommutative

analog of  orders in commutative rings. Examples of such algebras

include generalized Weyl algebras, the universal enveloping algebra of

the general linear Lie algebra and its deformations. The talk is based

on joint results with S.Ovsienko.

Jan 31 – Stefan Friedl (UQAM) “Subgroup separability and symplectic 4-manifolds”
In 1976 Thurston showed that if N is a fibered 3-manifold, then S^1 x N is symplectic. In this talk we will show that the converse holds if pi_1(N) satisfies certain subgroup separability properties. We will not assume any knowledge of symplectic geometry.

Feb 7 – Ben Steinberg (Carleton) Linear programming in right-angled Artin groups”
The classical linear programming problem is really the membership problem in finitely generated submonoids of free abelian groups.  So the membership problem for finitely generated submonoids of right-angled Artin groups can be viewed as a partially commutative analogue of linear programming.

In joint work with Markus Lohrey, we have classified which right-angled Artin groups having decidable membership in finitely generated submonoids. In the process we have obtained the first example, to our knowledge, of a finitely presented group with decidable generalized word problem, but undecidable membership in finitely generated submonoids.

The proof uses formal language theoretic techniques such as rational subsets, semilinear sets, context-free grammars and Parikh's theorem.

Feb 14 – No meeting

Feb 21 –? Study Break

Feb 28 – Peter Brinkmann (CCNY) Algorithmic aspects of free group automorphisms

I will present a survey of algorithms for free group automorphisms

and their mapping tori, with applications to surface homeomorphisms

and 3-manifolds. Some of these algorithms are practical and have

useful implementations. The focus will be on algorithms that use

dynamic properties of free group automorphisms, including recent

work on decision problems in free-by-cyclic groups.

Mar 7 – Pavel Zalesski (UnB) “The congruence subgroup problem: profinite aspect”
We begin with a detailed formulation of the congruence subgroup problem for the group SL_2(Z) and its negative solution discovered by Fricke and Klein. We shall then discuss how the congruence subgroup problem generalizes to arbitrary linear groups over arithmetic rings. After that we shall review the first positive results in the congruence subgroup problem obtaned by Bass-Lazard-Serre, Mennicke and Bass-Milnor-Serre in the late 60s. The centerpiece of this lecture will be a detailed discussion of the notion of congruence kernel introduced by Serre. We will explain why nowadays by the congruence subgroup problem people mean the problem of computation of the congruence kernel. We shall discuss then the congruence kernel for SL_2 and present  new results on its description  for some arithmetic lattices of the algebraic groups of rank 1.

Mar 21 – Daniel Wise (McGill) “An introduction to one-relator groups”

Groups with a presentation having a single defining relation have a long history in combinatorial group theory.

They have provided many interesting examples and led to various generalizations, but their theory is rather incomplete,

and has not been clarified by Gromov’s geometric approach to group theory.

I will give a brief survey of the theory of one-relator groups, focusing especially on Magnus’s method

and possibly on the parallel geometric approach using towers.

Mar 28 – Jitendra Bajpai (McGill) “Omnipotence of Surface Groups"
A group G is "potent" if for each nontrivial element g in G, and each natural number n,
there exists a finite quotient f:G -> Q, such that f(g) has order n.
Roughly speaking, the "omnipotent" is a multiple element generalization of "potent" which seeks to independently control the orders
of finitely many elements of G in a finite quotient. It is known that free groups are omnipotent.
In my talk I will explain why hyperbolic surface groups are omnipotent.

Apr 4 – Denis Osin (CCNY) Rips construction and Kazdan property (T)”
This is a joint work with Igor Belegradek. Applying Olshanskii's small
cancellation theory over hyperbolic groups we obtain the following
variant of the Rips construction. For any non--elementary hyperbolic
group $H$ and any finitely presented group $Q$, there exists a short
exact sequence $1\to N\to G\to Q\to 1$, where $G$ is a hyperbolic
group and $N$ is a quotient group of $H$.

This result is applied to construct a hyperbolic group that has the
same $n$--dimensional complex representations as a given finitely
generated group, to show that adding relations of the form $x^n=1$ to
a presentation of a hyperbolic group may drastically change the group
even in case $n>> 1$,  to prove that some properties (e.g. properties
(T) and FA) are not recursively recognizable in the class of
hyperbolic groups, etc. A relatively hyperbolic version of our theorem
can be used to generalize results of Ollivier--Wise on outer
automorphism groups of Kazhdan groups.

May 30 - Laszlo Babai (University of Chicago) Product-free sets according to Tim Gowers

Two decades ago, in a paper with Vera Sos, I asked a combinatorial question in finite groups.   Tim Gowers recently gave an elegant solution, connecting the question to the theory of quasirandomness in graph theory and to group representations.

The question concerns the following invariant.  A subset S of a group G is "product-free" if the equation xy=z has no solution in S.  Let a(G) denote the size of the largest product-free set in G.   We asked, is the quotient |G|/a(G) bounded?   We suggested the alternating groups as test cases.   Gowers gives a strong answer by showing that |G|/a(G) > m^{1/3} where m is the minimal dimension of nontrivial representations of G.   Gowers shows, in fact, that if X, Y, Z are subsets of G such that |X||Y||Z| > |G|^3/m  then the equation xy = z  has a solution such that  x belongs to X, y to Y, and z to Z.

I will present Gowers' elementary proof in full and indicate generalizations and applications to the theory of bounded generation obtained by Pyber, Nikolov, and the speaker.