


Combinatorial and Geometric Group Theory / Théorie des groupes combinatoire et géométrique Org: Inna Bumagin (Carleton) and/et Dani Wise (McGill)
 GREGORY BELL, Penn State University, University Park, PA 16802, USA
Growth of asymptotic dimension of groups

For a metric space X we consider an asymptotic dimension function
ad_{X} (l) which gives the dimension of X at the
"lscale". As l®¥ this function
gives the asymptotic dimension (asdim) of X. The growth of this
function is a quasiisometry invariant, and so its growth is
welldefined for finitely generated groups. Examining the growth of
ad_{G} (l) allows us to consider a notion of dimension for
groups with infinite asymptotic dimension. In this paper we examine
the growth of ad_{G} (l) for groups acting on finite
dimensional metric spaces. As a consequence we obtain results for
amalgamated products and HNN extensions, among others.
 BOB BURNS, York
Research of Yuri Medvedev at York, 19951999

Results of Yuri Medvedev from 19961999, all involving applications
of Lie rings to group theory, will be surveyed.
 RUTH CHARNEY, Brandeis University, Waltham, MA 02454, USA
Automorphism groups of spherical and affine type Artin Groups

We use results of Ivanov and Korkmaz on mapping class groups of
surfaces to determine the automorphism groups and abstract
commensurators of several of the finite and affine type Artin groups.
Joint work with John Crip.
 INDIRA CHATTERJEE, Cornell University
Discrete groups, operator norms and distortion

We shall discuss subgroup distortion, and explain how certain
geometric properties of a Cayley graph can give an obstruction to
having exponentially distorted subgroups.
 SEAN CLEARY, The City College of New York
A finitely presented group with unbounded deadend depth

An element g in a group G is a dead end with respect to a
generating set X if it is not adjacent to longer words in the Cayley
graph of G with respect to X. It is a "dead end" in the sense
that a geodesic ray from the identity to g cannot be extended beyond
g. Dead end elements come in different levels of severity, measured
by "deadend depth". In hyperbolic groups, there is always a global
bound on the depth of deadend elements, and Cleary and Taback showed
that the deadend depth of lamplighter groups is unbounded with
respect to standard generating sets. The lamplighter examples are not
finitely presentable. Here, we find a finitely presentable group
where there is no upper bound on the deadend depth of elements,
modelled on Baumslag's construction of a finitely presented metabelian
group containing Z wreath Z.
This is joint work with Tim Riley.
 ROSS GEOGHEGAN, SUNY at Binghamton
Cocompact proper \operatornameCAT(0) spaces

This is joint work with Pedro Ontaneda. We prove that every cocompact
proper CAT(0) space is "almost geodesically complete" aka
"almost extendible". A basic ingredient of the proof of this
geometric statement is the topological theorem that there is a top
dimension d in which the compactly supported integral cohomology of
X is nonzero. We also prove that the boundaryatinfinity (with
cone topology) has Lebesgue covering dimension d1. In all this we
do not assume that there is any discrete cocompact group of
isometries, not even a subgroup having discrete orbits; however, a
corollary for the discrete case case is that "the dimension of the
boundary" is a quasiisometry invariant of CAT(0) groups. (By
contrast, it is known that the topological type of the boundary is not
unique for a CAT(0) group.)
 YAIR GLASNER, UIC 851 S Morgan St., Chicago, IL 60607
Primitive groups, in various geometric settings

An action G×X ® X of a group on a set is
called primitive if there are no nontrivial G
equivariant equivalent relations on X. We say that a group G
is primitive if it admits some faithful primitive action on a
set. Primitive group actions can be thought of as the "irreducible
objects" in the theory of permutation groups.
In the spirit of the wellknown theorem of Margulis and Soifer, we
classify the finitely generated primitive groups in various geometric
settings. Including linear groups, hyperbolic groups, mapping class
groups and solvable groups.
In my talk I will describe the results and some applications. Time
permitting I will talk about some of the ideas that go into the
proof.
This is a joint work with Tsachik Gelander.
 BORIS GOLDFARB, University at Albany, SUNY, Albany, NY 12222
On the algebraic structure of geometric group rings

I will show how to use noncommutative localization to relate the
question of existence of zero divisors in group rings to existence of
certain Ktheoretic invariants which were the object of intense
study in recent years.
 JOINT PANEL DISCUSSION
Participating Sessions

This is a joint Panel Discussion of the following sessions:
 Combinatorial and Geometric Group Theory
 Groups, Equations, Noncommutative Algebraic Geometry
 Interactions between Algebra and Computer Science
 JOHN MEIER, Lafayette College
Euler characteristics of automorphism groups of free products

One of the basic questions in group cohomology is determining the
Euler characteristic of a group or interesting family of groups. We
find and exploit a combinatorial summation identity over the lattice
of labelled hypertrees that allows us to gain concrete information on
the Euler characteristics of various automorphism groups of free
products of groups. In particular, we establish formulae for the
Euler characteristics of: the group of Whitehead automorphisms Wh
(*_{i=1}^{n} G_{i}) when the G_{i} are of finite homological type;
Aut(*_{i=1}^{n} G_{i}) and Out(*_{i=1}^{n} G_{i}) when the G_{i}
are finite; and the palindromic automorphism groups of finite rank
free groups.
This is joint work with Craig Jensen and Jon McCammond.
 LUIS RIBES, Carleton University, Ottawa, ON K1S 5B6
Prop groups acting on profinite trees

This work describes the structure of countably generated prop
groups that act continuously on profinite trees under certain
conditions. As a consequence one obtains Kuroshtype results
describing closed prop subgroups of free products of profinite
groups, recovering in a unified manner results of Melnikov, Haran, and
RibesHerfort.
 TIM RILEY, Yale
Diameter and filling length of van Kampen diagrams

I will explain recent work with Martin Bridson on the diameter
and filling length of van Kampen diagrams and the corresponding
filling functions for finitely presented groups.
To measure the diameter of a van Kampen diagram D one uses the
path metric on the 1skeleton of either D itself or the Cayley
2complex it maps into. We give groups in which there are qualitative
differences between the two resulting diameter filling functions.
Filling length controls the length of the contracting loop in the
course of a nullhomotopy across a van Kampen diagram. We exhibit a
group in which the filling length of loops varies dramatically
depending on whether or not one keeps a base vertex fixed during the
nullhomotopy.
 YVES STALDER, Université de Neuchâtel, Institut de mathématiques, Rue
EmileArgand 11, CH2007 Neuchâtel, Suisse
Convergence of BaumslagSolitar groups

The set of marked groups on two generators can be endowed with a
natural topology, so that it becomes a compact Hausdorff totally
disconnected space. This framework allows, for instance, a
topological definition of Sela's limit groups.
BaumslagSolitar groups
BS(m,n) = áa,b  ab^{m} a^{1} = b^{n} ñ 

are canonically marked by the generators a,b. We are interested by
the limits of these groups. We will prove that lim_{m,n ® ¥}BS(m,n) is the free group on two generators and give some conditions
on the sequence (k_{n})_{n} such that lim_{n ® ¥}BS(m,k_{n}) exists. These conditions are related to padic
integers.
 BENJAMIN STEINBERG, Carleton University, 1125 Colonel By Drive, Ottawa ON K1S 5B6
On the spectra of lamplightertypes groups and Cayley Machines

Grigorchuk and Zuk calculated the spectrum and spectral measure of the
Markov operator associated to a simple random walk on the lamplighter
group Z/2Z\wr Z by realizing it as the
group generated by a twostate automaton. In particular, the spectral
measure was discretea new phenomenon for such Markov operators.
Dicks and Schick generalized their results to other wreath product
groups G\wr Z, with G a finite group, using a different
technique.
If G is a finite group, then the Cayley graph of G (with respect
to all generators) can be turned into a Mealy automaton, called the
Cayley Machine of G, generating in this way an automaton group. If
G is Abelian, one obtains the lamplightertype group G\wrZ, generalizing Grigorchuk and Zuk who studied the case G = Z/2Z. In general, the group of a Cayley machine is
locally finite by infinite cyclic, but is not isomorphic to a wreath
product of a finite group with a torsion group. The semigroup
generated by the states is free on G generators.
Using the techniques of Grigorchuk and Zuk, we compute the socalled
Kestenvon NeumannSerre spectral measures associated to the Schreier
graphs with respect to a parabolic subgroups of our groups and the
Ihara Zeta function of the infinite Schreier graphs. In the case of
G Abelian, we prove the Kestenvon NeumannSerre spectral measure is
the spectral measure associated to the random walk on G\wrZ, recovering the results of Dicks and Schick.
 JENNIFER TABACK, Bowdoin College, Brunswick, ME 04011
Geodesic languages for Lamplighter groups and Thompson's
group F

We show that both the lamplighter groups in the standard wreath
product presentation, and Thompson's group F in the standard finite
presentation have infinitely many cone types, and thus no regular
language of geodesics. We prove a general theorem about certain types
of elements which prohibit a group from having a regular language of
geodesics.
Additionally, I will show that each lamplighter group has a context
free language of geodesics with unique representatives for each
element, and a counter language with the same property.
This is joint work with Sean Cleary and Murray Elder.
 FRANCIS TANG, University of Waterloo, Waterloo, ON
Conjugacy separability of Seifert 3manifold groups over
nonorientable surfaces

Seifert 3manifold groups have nice separability properties. Niblo
showed that they are double coset separable. Allenby, Kim and Tang
showed that most of the Seifert 3manifold groups over over orientable
surfaces are conjugacy separable. In this talk we discuss the
conjugacy separability of Seifert 3manifold groups over
nonorientable surfaces. It turns out all Seifert 3manifold groups
over nonorientable surfaces are conjugacy separable. We conjecture
that all Seifert 3manifold groups are conjugacy separable.
 TED TURNER, University at Albany
Test ranks of groups and the Magnus problem FP15

In the Magnus problem list, problem FP15 is the following.
"Let G be an ngenerator group. Call a set of elements
{g_{1},...,g_{k}}, a test set for the group G if, whenever
f(g_{i}) = g_{i}, i = 1,...,k 

for some endomorphism f of G, this f is actually an automorphism
of G. The test rank of G is the minimal cardinality of a
test set. Can the test rank be equal to 2 if n > 2?" (Posed by Ben
Fine.)
I will describe a positive solution. In particular, I will describe
how to compute the test rank of any finitely generated abelian group
and show that all possible combinations of rank and test rank are
realized.
This is joint work with Charles F. Rocca Jr.
 KAREN VOGTMANN, Cornell University
Cycles at infinity in Outer space

Using a variation of Bestvina and Feighn's bordification of Outer
space, we produce a chain complex which computes the rational
cohomology of the group Out(F_{n}) of outer automorphisms of a free
group. We show how to produce a great many cycles in this chain
complex, including cycles which give nontrivial cohomology classes in
H^{4} ( Out(F_{4});Q ) and H^{8} ( Out(F_{6});Q ).
We conjecture that these cycles span all of the rational cohomology of
Out(F_{n}). The construction of these cycles was inspired by a
graphical interpretation of certain cycles, constructed by S. Morita,
in a different chain complex which also computes the cohomology of
Out(F_{n}).
This is joint work with Jim Conant.
 KEVIN WORTMAN, Cornell University, Department of Mathematics, Ithaca, NY
14853, USA
Finiteness properties of arithmetic groups over function
fields

In this talk we'll focus on a proof of Nagao's theorem that SL_{2}(F_{q}[t]) is not finitely generated. Here F_{q} [t]
is a ring of polynomials with one variable and coefficients in a
finite field.
The proof presented will not be new, but I'll explain how it arises as
a special case of a new proof used by KaiUwe Bux and myself to verify
the conjecture that an arithmetic subgroup of a reductive group
G defined over a global function field K is of type
FP_{¥} if and only if the semisimple Krank of G
equals 0.
This conjecture had its roots in the work of Serre and Stuhler. Our
proof is motivated by the EpsteinThurston proof that SL_{n}(Z) is not combable when n ³ 3.

