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)
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)
“Around
Whitehead's algorithm”
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
of Whitehead's algorithm traditionally regarded as
"slow" may not be
that slow after all. This conjecture was confirmed
by B.Khan for
the free group of rank 2 and reinforced (although
not yet confirmed)
for free groups of any finite rank by D.Lee.
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 (
"The conjugacy problem for free-by-free
groups"
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
problem
for free-by-cyclic groups. The proof is rather simple, after using
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 (
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 (
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 (
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.
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.
