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

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

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

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

** **

**Winter 2006:**

** **

January 9 – Sergei Ivanov (UIUC) (3pm Room 1234 Burnside)

“On the Kurosh rank of
subgroups in free products of groups”

We will
discuss the Kurosh rank of the intersection of subgroups in free products of
groups

** **

January 18 – Denis Serbin (McGill)

“Groups with free regular length functions in Z^n”

Finding
description of finitely generated groups acting freely on Lambda-trees, or,
equivalently

groups
having free length functions in Lambda (where Lambda is an ordered abelian
group) is one of the major problems in Geometric Group Theory. This problem was
solved for some special cases of Lambda but still is far from being solved in
the general case. In my talk at first I'm going to introduce a natural
restriction on length function which is called the regularity condition. Then
I'm going to present the description of finitely generated groups with regular
free length functions in Z^n, and to discuss why regularity condition is
important.

This
is a joint work with O. Kharlampovich, A. Myasnikov and V. Remeslennikov."

January 25 –
Olga Kharlampovich (McGill)

*“Equations with parameters in a free
group”*

Abstract: We will discuss algebraic, transcendental
and reducing solutions

of equations with parameters in a free group.

We will also discuss differences and similarities
with commutative algebra.

** **

Feb 1 – Alexei
Miasnikov (McGill)

“Asymptotic group theory and
cryptography”

In this talk I am going to discuss some recent
applications of asymptotic

group theory to cryptanalysis of various
non-commutative cryptosystems. In

particular, I will try to explain effectiveness of
some well-known heuristic attacks

and show how one can adjust crypto schemes to avoid
these attacks.

** **

Feb 8 – Joe
Lauer (McGill)

“Cubulating
one-relator groups with torsion”

In his
thesis Sageev showed that given a group G with a codimension 1

subgroup one can construct an action of G on a
CAT(0) cube complex. In

certain situations a variant of his construction
can be used using

codimension-1 subpaces of the universal cover. Proving geometric

properties about these subspaces then leads to nice
properties of the

action. In
this talk I will discuss the construction for one-relator

groups with torsion.

Feb 15 –
Joseph Maher (UQAM) “Random walks on the mapping class group”

We show that
the probability that a random walk of length n on the mapping class group of a
closed orientable surface is pseudo-Anosov tends to one as n tends to infinity.
We use this to show that if the random walk is used as the gluing map for a
Heegaard splitting, then the probability you get a hyperbolic manifold tends to
one as n tends to infinity.

(Thursday, Feb 16, 11:00-12:00 1234 Burnside Hall)

Lior Silberman (IAS)

*"Poincare
Inequalities and Fixed Points"*

Abstract: Combining several existing results in metric
geometry we show

that
the wild group constructed by Gromov has a strong fixed point

property
for isometric actions, generalizing the proof that the group has

Kazhdan property (T). This is
joint work with Assaf Naor.

Feb
22 – Break!?

** **

March 1 - Jane
Gilman (

*“**Informative Words and Discreteness
Criteria”
*There are certain
families of words and word sequences (words in the

generators of a two-generator group) that arise frequently in the Te-

ichmuller theory of hyperbolic three-manifolds and Kleinian and Fuch-

sian groups and in the discreteness problem for two generator matrix

groups. We survey some of the families of such words and sequences:

the semigroup of so called good words of Gehring-Martin, the so called

killer words of Gabai-Meyerhoff-NThurston, the Farey words of Keen-

Series and Minsky, the discreteness-algorithm Fibonacci sequences of

Gilman-Jiang and parabolic dust words. We report on connections

between these families.

March 8– Mark
Sapir (Vanderbilt)

*“Groups acting
on tree-graded spaces and homomorphisms into relatively hyperbolic
groups”*

This is a joint work with
Cornelia Drutu. Tree-graded spaces are generalizations of R-trees. They appear
as asymptotic cones of groups (when the cones have cut points). Since many
questions about endomorphisms and automorphisms of groups, solving equations
over groups, studying embeddings of a group into another group, etc. lead to
actions of groups on the asymptotic cones, it is natural to consider actions of
groups on tree-graded spaces. We develop a theory of such actions which
generalizes the well known theory of groups acting on R-trees. As applications
of our theory, we describe, in particular, relatively hyperbolic groups with
infinite groups of outer automorphisms, and co-Hopfian relatively hyperbolic groups.
See http://front.math.ucdavis.edu/math.GR/0601305
.

March 15 –
Daniel Wise (McGill) *“Subgroup
separability of prime alternating link groups"
*The link L is alternating if circles alternately pass above and below crossing
points of a planar projection. The link L is prime if it is not the connected sum of two nontrivial links. A
group is subgroup separable if
every finitely generated subgroup is the intersection of finite index subgroups.
Let G be the fundamental group of the complement of a prime alternating link:

Theorem 1: G is subgroup separable.

Theorem 2: G is virtually a subgroup of a right-angled Artin group.

Corollary 3: G is a subgroup of SL(n,Z) for some n.

March 22 – Kim
Ruane (Tufts) “*A Converse to a Splitting Theorem”*

In previous
joint work, M. Mihalik and I showed that certain amalgamated product splittings
(and HNN extensions) of a one-ended CAT(0) group force the boundary of the
space on which the group is acting to have points of non-local
connectivity. In this talk, we discuss a converse to this theorem in the
setting of groups acting on CAT(0) spaces with the Isolated Flats Property.

March 29 –
Ekaterina Blagoveshchenskaya (

*“Almost completely decomposable groups
and rings”*

The
theory of Almost Completely Decomposable Groups (*acd groups*) has been intensively developed during the last decades.
According to the definition, any acd group is a torsion-free abelian group of
finite rank which has a completely decomposable subgroup of finite index. Then such groups form the class which is the
closest one to the completely decomposable group class (consisting of finite
direct sums of rank-one groups). However, acd group properties tightly
connected with their endomorphism ring characteristics are very different from
those of completely decomposable groups. The main basis for combining group and
ring approaches in case of acd groups is the fact that End(*X*) of an acd group *X* is
again an acd group with respect to the sum operation.

The
acd groups and rings are considered in the dual connection between them. The
classification, realization and near-isomorphism theorems are obtained for some
classes of rings with acd additive structures. The approach is generalized to
the so-called local acd groups and rings of infinite ranks.

April 5 – Inna
Bumagin (Carleton) * **“Homomorphisms to relatively hyperbolic
groups”*

Given a
finitely generated group A and a relatively hyperbolic group G, how much are
properties of Hom(A,G) and of A itself related? In

this talk, I
will survey the known results and applicable techniques.

April 12 - Jitendra Bajpai (McGill) *“Omnipotent
Groups”*

April 19 – Ben Steinberg (Carleton) *“On rational subsets of groups”*

A rational subset of a group G is a subset that can be built
up from the finite subsets by applying finitely many times the operations:
union, product and generating submonoids. Equivalently, these are the
subsets of groups recognized by finite state automata.

A group has decidable rational subset problem if membership
in rational subsets is decidable. This implies a decidable word problem,
generalized word problem and submonoid membership problem. Also the
finite order problem is decidable for such groups. Known examples include
virtually free groups and virtually abelian groups. In this talk we show
that a finite graph of groups with vertex groups having decidable rational
subset problem and finite edge groups again has decidable rational subset
problem. We also show that a direct product of a free group with an
abelian group has decidable rational subset problem. Finally we give a
transparent proof of Grunschlag's result that the decidability of the rational
subset problem is a virtual property. This is joint work with M. Kambites
and P. Silva.

April 26 – Marc Bourdon (Lille) *“Lp-cohomology for finitely generated groups”*

l_p-cohomology
is a quasi-isometry invariant for finitely generated groups. The talk will
explain

some basic
results mainly issued from Gromov's book "Asymtotic invariants for
infinite groups".

** **

**Fall 2005:**

Sept 28 – Christophe Reutenauer (UQAM)

*“Christoffel words and automorphisms of the
free group on two generators”*

Christoffel words are a
finitary version of sturmian sequences. These are related to continued
fractions, discretization of lines in
the plane and to combinatorics on words and palindromes, and are studied since Bernouilli, Smith,
Christoffel, Markoff, Morse, Hedlund and
many others at the end of the 20th century. We give some elements of this
theory, by stressing the geometric approach. We show how these words allow to
construct all bases of the free group F2. We
show how the sturmian morphisms (substitutions which preserve
sturmian sequences) lead to a structure
theorem of the group of automorphism of
F2, that uses the braid group on 4 strands.

Sept 21 – Daniel Wise (McGill) *“Quasiconvex
subgroups have finite height”*

The *height* of a
subgroup *H* in *G* is the maximal number of distinct conjugates of *H*
whose total intersection is infinite.
The *width* of *H* is the maximal number of distinct
conjugates of *H *whose pairwise intersection is infinite. It was proven by Gitik-Mitra-Rips-Sageev that
any quasiconvex subgroup of a word-hyperbolic group has finite width, and hence
finite height. I will discuss a proof of this result aimed towards a
generalization.

This is joint work with Chris Hruska.