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 (
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 congruence kernel introduced by Serre.
We will explain why nowadays by the congruence subgroup problem people mean the
problem of computation of the congruence kernel. We shall discuss then the
congruence kernel for SL_2 and present new results on its
description for some arithmetic lattices of the algebraic groups of rank
1.
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 G is
"potent" if for each nontrivial element g in G, and
each natural number n,
there exists a finite quotient f:G
-> Q, such that f(g) has order n.
Roughly speaking, the "omnipotent" is a multiple element
generalization of "potent" which seeks to independently control the
orders
of finitely many elements of G in a finite quotient. It is known that
free groups are omnipotent.
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 (
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.