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 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 (Rutgers)

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 .


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 (St Petersburg Polytechnical University)

“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.