The seminar meets each Wednesday at 3:30pm in
920 Burnside Hall at 805

**(Here
is a link to our schedule during 2014-2015)**

**">(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)**

**(Here is a link to
our schedule during 2003-2004) **

**(Here is
a link to our schedule during 2002-2003) **

**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 (*“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

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

of finitely many elements of

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 (*“**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.