Geometry/Topology Seminar
Fall 2007
Thursdays (and sometimes Tuesdays) 23pm, in
Ryerson 358

 Thursday October 4 at 2pm in Ry358
 Dieter Kotschick, University of Munich
 Stable length in stable groups

Abstract: We show that the stable commutator length
vanishes for certain groups defined as infinite unions of
smaller groups. The argument uses a grouptheoretic analogue
of the Mazur swindle, going back to the works of Anderson,
Fisher, and Mather on homeomorphism groups. Groups to which
this applies include the stable mapping class group, the
stable automorphism group of free groups, the braid group on
infinitely many strands, and various diffeomorphism groups.

 Thursday October 18 at 2pm in Ry358
 Alex Furman, University of Illinois at Chicago
 Invariant and stationary measures for groups of
toral automorphisms

Abstract: Joint work with J. Bourgain, E.
Lindenstrauss, S. Mozes. Given a Zariski dense group
G of toral automorphisms we prove that the only
invariant or, more generally, stationary measures on the
torus are combinations of Lebesgue and atomic measures.

 Tuesday October 30 at 2pm in Ry358
 Mahan Mj, RKM Vivekananda University
 Local Connectivity of Limit Sets and CannonThurston Maps

Abstract: We shall briefly sketch the ideas that go
into two related theorems:
 Connected limit sets of
(3 dimensional) finitely generated Kleinian groups are
locally connected
 Such groups admit a continuous map
from the Gromov boundary to the limit set, extending the
natural (inclusion) map of the Cayley graph into the
universal cover.

 Thursday November 1 at 2pm in Ry358
 Rupert Venzke, Caltech
 Braid forcing, hyperbolic geometry, and pseudoAnosov sequences of low entropy

Abstract: We view braids as automorphisms of
punctured disks and define a partial order on pseudoAnosov
braids called the “forcing order”. The order
measures whether one automorphism induces another given
automorphism on the surface. PseudoAnosov growth rate
decreases relative to the order and appears to give a good
measure of braid complexity. Unfortunately it appears
difficult computationally to determine explicitly the
partial order structure by hand. We use several computer
algorithms to study the bottom part of the partial order
when the number of braid strands is fixed. From the
algorithms, we build sequences of low entropy pseudoAnosov
nstrand braids that are minimal in the sense
that they do not force any other pseudoAnosov braids on the
same number of strands. The sequences are an extension of
work done by Hironaka and Kin, and we conjecture the
sequences to achieve minimal entropy among certain
nontrivial classes of braids. In general, the lowest entropy
pseudoAnosov braids appear to have mapping tori that come
from Dehn surgery on very low volume hyperbolic 3manifolds
and we begin to analyze the relation between entropy and
hyperbolic volume.

 Thursday November 15 at 2pm in Ry358
 Marco Varisco, Binghamton University
 Isomorphism conjectures in algebraic Ktheory and topological
cyclic homology

Abstract: Algebraic Ktheory of group rings plays a
major role in geometric topology, in classification problems
for manifolds and their automorphisms, but it is usually
very hard to calculate. The socalled isomorphism conjecture
of Farrell and Jones predicts that the Kgroups of group
rings are isomorphic to the (equivariant, generalized)
homology groups of certain universal spaces, which are much
more amenable to computations. I will explain and motivate
this conjecture and describe how to obtain partial results
using trace invariants of Ktheory taking values in
topological cyclic homology. A connection with (Schneider's
generalization of) the Leopoldt conjecture in algebraic
number theory will also be highlighted.

 Thursday November 29 at 2pm in Ry358
 Dimitri Shlyakhtenko, UCLA
 Free entropy dimension for groups

Abstract: Free entropy dimension is a quantity
introduced by D. Voiculescu in the context of his free
probability theory. In the case of a group, this number is,
very roughly, a (renormalized) asymptotic dimension of the
set of unitary representations of the group into an
ultraproduct of finitedimensional algebras. This number
turns out to be connected with the first
L^{2} Betti number of the group. While an
estimate one way always holds, we will discuss more recent
work on proving the other estimate in certain circumstances,
using ideas from free stochastic calculus.

 Thursday December 6 at 2pm in Ry358
 Peter Constantin, University of Chicago
 Nonlinear FokkerPlanck Equations

Abstract: Nonlinear FokkerPlanck equations describe
the probability distributions of particles diffusing on
manifolds. The manifolds are configuration spaces of the
particles: these can be smooth compact Riemannian manifolds,
but also more complicated objects. I will discuss limit
distributions when the particle interaction becomes strong.
The talk will be accessible to second year graduate
students.
For questions, contact