Montreal Geometric & Combinatorial Group Theory Seminar

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

Winter 2005:

Apr 6 –

Mar 30 – Reserved

Mar 23 – Daryl Cooper (UCSB) "Is there a Universal Geometry for 3-Manifolds ?"

Abstract: We will discuss real projective geometry and the

new notion of pseudo-conformal geometry, both in

relation to 3-manifolds. Groups are involved !

Mar 16 - Vladimir Shpilrain (CCNY)

Abstract: The interest to Whitehead's algorithm for solving the automorphic

conjugacy problem in a free group was revived a few years ago after a work of

A.G.Myasnikov and Shpilrain made it appear likely that the part

that slow after all. This conjecture was confirmed by B.Khan for

the free group of rank 2 and reinforced (although not yet confirmed)

On the other hand, the time complexity of the algorithm as a function

of the rank of the ambient free group remains very high (at least, in theory).

We shall discuss possible remedies.

On the third(??) hand, the study of the generic-case complexity of

Whitehead's algorithm (joint work with I.Kapovich and P.Schupp) gives rise

to some interesting questions about statistical properties of elements

of a free group; we are going to discuss these as well.

Mar 9 – Dani Wise (McGill) “Cubulating Groups”

Abstract: Since their introduction by Gromov nearly 20 years ago, CAT(0) cube complexes have continued to play an increasingly prominent role in geometric group theory. This talk will survey both well-known results and recent developments in the area.

Mar 2 – Olga Kharlampovich (McGill) "Decision algorithm for EA-sentences in a free group".

Abstract: We describe an algorithm that determines whether a given

EA-sentence is true in a free group (of any rank greater than one). This

is a partial case of a solution (joint with A. Miasnikov) of the Tarski

problem about the decidability of the elementary theory of a free group.

Feb 23 – Break

Feb 16 – CANCELLED

Feb 9 – Denis Serbin (McGill)

"Merzlyakov's theorem for groups with free regular length functions"

Abstract: A first-order sentence is called positive

if it contains no negations or implications. It was

proved by Yu.Merzlyakov that a positive sentence true

in a free group F can be reduced to a positive

universal sentence (all quantifiers are universal).

Moreover, Merzlyakov proved existence of Skolem

functions for positive theory of F.

We introduce an analog of Merzlyakov's result for the

class of groups with free regular length functions.

This is joint work with Bilal Khan and Alexei G.

Miasnikov.

Feb 2 - Alexei Myasnikov (McGill)

"Random van Kampen diagrams and algorithmic problems in groups"

Abstract: In this talk we are going to discuss the structure of

random van Kampen diagrams over finitely presented groups. Such

diagrams have many remarkable properties which allow one to design

new fast algorithms for the classical algorithmic  problems in

groups. In particular, we will show that  the generic  case time

complexity of the search word problem in finitely presented groups

is polynomial. (joint with  Sasha Ushakov)

Jan 26 - Dani Wise (McGill)

"The Tits Alternative for groups acting on cube complexes"

Abstract: In 1972, J. Tits proved that a linear group G has the following property:

For every finitely generated subgroup of H of G, either H contains a rank 2 free group

or H is virtually solvable. This property has come to be called "The Tits Alternative",

and has since been proven for various classes of tractable groups.

In my talk, I will prove the Tits alternative for groups acting freely on finite dimensional

CAT(0) cube complexes. This is joint work with Michah Sageev.

Jan 19 - E. Ventura (Barcelona)

Abstract: (joint work with A. Martino and O. Bogopolski). In the first

part of this talk, I will give a positive solution to the conjugacy

two deep results about free groups (although the algorithm provided is in

fact very complicated). In the second part of the talk these techniques

will be used to study the conjugacy problem in a bigger family of groups,

namely extensions of free and free abelian groups by torsion-free

hyperbolic groups (so, including free-by-free groups). This bigger family

is more interesting because it contains groups with unsolvable conjugacy

problem. Our main result in this direction is a characterization of those

groups in the family which have solvable conjugacy problem, in terms of

what we call "orbit decidability".

Jan- 12  Ilya Kazachkov (Russian Academy of Science)

A Gathering Process in Artin Braid Groups

Abstract:

In this talk I shall construct a gathering process by the means of which I

obtain new normal forms in braid groups. The new normal forms generalise

Artin-Markoff normal forms and possess an extremely natural geometric

description. Then I plan to discuss the implementation of the introduced

gathering process, to derive some interesting corollaries and, in particular,

offer a method of generating a random braid.

Fall 2004:

Aug 31-  (Tuesday) Alain Valette, (Neuchâtel)

CRITICAL EXPONENTS AND THE FIRST L^p-COHOMOLOGY

Abstract:  For a countable group $\Gamma$ acting isometricaly on a metric space $X$, the critical exponent $e(\Gamma)$ measures the rate of growth of the intersection of $\Gamma$-orbits with balls in $X$. We will present the following rigidity result, obtained jointly with M. Bourdon and F. Martin: let $\Gamma$ be a co-compact lattice in the isometry group of a rank 1 Riemannian symmetric space, and let $\Lambda$ be a group acting properly discontinuously, isometrically, on a CAT(-1) space; if $\Gamma$ is isomorphic to $\Lambda$, then $e(\Gamma)\leq e(\Lambda)$. The proof rests on the study of the first $L^p$-cohomology of $\Gamma$.

Sept 1- Alain Valette, (Neuchâtel)

SPACES WITH MEASURED WALLS, PROPERTY (T) AND THE HAAGERUP PROPERTY

Abstract:  Spaces with walls are combinatorial objects providing one of the most efficient ways to construct affine isometric group actions on Hilbert space. If the group has Kazhdan's property (T), this yields fixed points theorems for group actions on spaces with walls. On the other hand, if the group action on the space with walls is proper, so is the corresponding affine action, i.e. the group has the Haagerup property (or is a-T-menable in Gromov's terminology). Consideration of less combinatorial situations (i.e. real hyperbolic spaces, real trees...) leads to defining spaces with measured walls. We conjecture that a locally compact group has the Haagerup property if and only if it admits a proper action on a space with measured walls, and prove this conjecture for some special classes, e.g. discrete groups.

Sept 22- Ilya Kapovich (UIUC)

Translation equivalence in free groups

Abstract:           We discuss the pheonomenon when two elements of a free group

have equal translation lengths with respect to every free isometric action

of this group on a real tree. We give a combinatorial characterization of

this phenomenon, called translation equivalence, in terms of Whitehead

graphs and exhibit two different sources of it. The first source of

translation equivalence comes from representation theory and the so-called

SL_2 "trace identities" or "Fricke characters". The second source comes

from geometric properties of groups acting on real trees and a certain

power redistribution trick. We also analyze to what extent these are

applicable to the tree actions of surface groups that occur in the

Thurston compactification of the Teichmuller space. This is joint work

with Gilbert Levitt, Paul Schupp and Vladimir Shpilrain.

Sept 24- Friday 4pm, UQAM Colloquium

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

Paul Schupp (UIUC)

# Computational Complexity and Geometric Group Theory

Abstract:          Decision problems and algorithms have been a major part of group theory since Dehn formulated the word problem in 1912. We now realize that the area interacts with the theory of computational complexity in important ways. Current questions of exactly what a good measure of "complexity" is in certain situations are also important. This talk will discuss some recent results relating geometric group theory to complexity theory.

Oct 4 Monday 3:30pm, room 920– Boris Plotkin (Hebrew University)

Automorphisms of categories of free algebras of varieties: Applications

Abstract:         We consider general methods of investigation of automorphisms of

categories of free algebras of varieties. These general methods are used in

the categories of free groups, free semigroups, free associative and commutative

algebras, free associative algebras, free Lie algebras and the category of free

representations of groups. Further this theory is applied to algebraic geometry

in different varieties of algebras. Special attention is paid to noncommutative algebraic

geometry and algebraic geometry in group representations.

Oct 6 - Richard Weidmann (Frankfurt)

Title: Accessibility of finitely generated groups and the rank problem

Abstract:          A group G is called accessible if there is a bound on the complexity of a splitting of G as the fundamental group of a graph of groups with finite edge groups. Dunwoody showed that all finitely presented groups are accessible but also constructed a non-accessible finitely generated group.

In this talk we discuss various accessibility results for finitely generated groups, in particular we discuss Linell accessibility, Sela's acylindrical accessibility and its generalizations. We further show that in some situations where accessibility holds a relationship can be established between the rank of the fundamental group of a graph of groups and the rank of the vertex and edge groups.

If time permits we discuss how the techniques that are used to prove the above results give a solution to the rank problem for some classes of groups, in particular for Fuchsian groups and word-hyperbolic, torsion-free Kleinian groups.

Oct 20 - Chris Hruska (U. Chicago)

Title:   "Commensurability invariants of nonuniform tree lattices"

Abstract:          If X is a locally finite tree, its group of automorphisms  G=Aut(X) is a locally compact group. A lattice in G is a discrete

subgroup with cofinite Haar measure. With the right normalization of Haar  measure, there is a simple combinatorial formula for the Haar measure, or covolume'' of a lattice. A study of these tree lattices'' generalizes the study of lattices in Lie groups over a nonarchimedean local field, and provides a remarkably rich theory (see the recent book by Bass-Lubotzky).One of the basic problems about a locally compact group is to classify its lattices up to commensurability. Outside the setting of linear groups, commensurability invariants have been hard to come by. We introduce two new commensurability invariants, and construct lattices realizing every possible choice of these invariants. In particular, we construct uncountably many noncommensurable lattices with any given covolume.(This is joint work with Benson Farb)

Nov 17- Dani Wise (McGill)

Title: “Hopfian and Non-Hopfian Groups”

Abstract: A group G is "Hopfian" if every surjective endomorphism of G is an automorphism.

I will survey the known results and examples relating  to Hopf's property.

Nov 24 – Lysionok (Steklov Institute)

Title: “Conjugation chains and groups with Burnside structure of finite subgroups”

We study the following property of an infinite group $G$: There is a number

$\ell>0$ such that for any $x,w\in G$, if $x$, $w^{-1}xw$, $\dots$, $w^{-\ell}xw^\ell$

generate a finite subgroup $H$ then $w$ lies in the normalizer of $H$. This property

plays an essential role in all known approaches to Burnside groups of sufficiently large

exponents. We formulate an algebraic sufficient condition for a group $G$ to have this

property, which extracts a specific algebraic part of the approach to Burnside groups.