Geometry/Topology Seminar
Fall 2016
Thursdays (and sometimes Tuesdays) 34pm, in
Eckhart 308

 Thursday September 22 at 34pm in Eck 308
 Sebastian Hensel, Bonn
 Rigidity and Flexibility for the Handlebody Group

Abstract: The handlebody group H_{g} is the
subgroup of the mapping class group Mod_{g} of a
surface formed by all those elements which extend to a given
handlebody. In this talk we will first show that finite
index subgroups of this group are rigid: any inclusion into
Mod_{g} is conjugate to the standard inclusion. We
then discuss flexible behaviour: the existence of inclusion
of H_{g} into Mod_{h} whose image is not
conjugate into any handlebody subgroup of Mod_{h}.

 Thursday September 29 at 34pm in Eck 308
 Jesse Wolfson, University of Chicago
 Coincidences of homological densities, predicted by arithmetic

Abstract: Basic questions in analytic number theory
concern the density of one set (e.g. squarefree integers)
in another (e.g. all integers). Motivated by Weil's number
field/function field dictionary, we introduce several
topological analogues, measuring the “homological
density” of one space in another. In arithmetic, Euler
products can be used to show that many seemingly different
densities coincide in the limit. By combining methods from
manifold topology and algebraic combinatorics, we discover
analogous coincidences for limiting homological densities
arising from spaces of 0cycles (e.g. configuration spaces
of points) on smooth manifolds and complex varieties. We do
not yet understand why these topological coincidences occur.
This is joint work with Benson Farb and Melanie Wood.

 Thursday October 6 at 34pm in Eck 308
 Arie Levit, Hebrew University of Jerusalem
 Local rigidity of uniform lattices

Abstract: A lattice is topologically locally rigid
(t.l.r) if small deformations of it are isomorphic lattices.
Uniform lattices in Lie groups were shown to be t.l.r by
Weil [60']. We show that uniform lattices are t.l.r in any
compactly generated topological group. A lattice is locally
rigid (l.r) is small deformations arise from conjugation. It
is a classical fact due to Weil [62'] that lattices in
semisimple Lie groups are l.r. Relying on our t.l.r results
and on recent work by CapraceMonod we prove l.r for uniform
lattices in the isometry groups of proper geodesically
complete CAT(0) spaces, with the exception of SL2(R) factors
which occurs already in the classical case. Moreover we are
able to extend certain finiteness results due to Wang to
this more general context of CAT(0) groups. In the talk I
will explain the above notions and results, and present some
ideas from the proofs. This is a joint work with Tsachik
Gelander.

 Thursday October 13 at 34pm in Eck 308
 Rita Gitik, University of Michigan
 ON INTERSECTIONS OF CONJUGATE SUBGROUPS

Abstract: We define a new invariant of a conjugacy
class of subgroups which we call the weak width and prove
that a quasiconvex subgroup of a negatively curved group has
finite weak width in the ambient group. Utilizing the coset
graph and the geodesic core of a subgroup we give an
explicit algorithm for constructing a finite generating set
for an intersection of a quasiconvex sub group of a
negatively curved group with a conjugate. Using that
algorithm we construct algorithms for computing the weak
width, the width and the height of a quasiconvex subgroup of
a negatively curved group. These algorithms decide if a
quasiconvex subgroup of a negatively curved group is almost
mal normal in the ambient group.

 Thursday October 20 at 34pm in Eck 308
 David Fisher, Indiana University
 Character varieties and actions on products of trees

Abstract: Can a surface group act freely and
properly on a finite product of bounded valence trees? I'll
discuss some motivation for this question, some variants and
generalizations and one attempt to provide an answer using
character varieties in characteristic p.

 Thursday November 03 at 34pm in Eck 308
 Andy Putman, Notre Dame
 The high dimensional cohomology of the moduli space of curves
with level structures

Abstract: I will prove that the moduli space of
curves with level structures has an enormous amount of
rational cohomology in its cohomological dimension. This is
joint work with Neil Fullarton.

 Thursday November 10 at 34pm in Eck 308
 Priyam Patel, University of California, Santa Barbara
 Algebraic and topological properties of big mapping class groups

Abstract: The mapping class group of a surface is
the group of homeomorphisms of the surface up to isotopy (a
natural equivalence). Mapping class groups of finite type
surfaces have been extensively studied and are, for the most
part, wellunderstood. There has been a recent surge in
studying surfaces of infinite type and in this talk, we
shift our focus to their mapping class groups, often called
big mapping class groups. In contrast to the finite type
case, there are many open questions regarding the basic
algebraic and topological properties of big mapping class
groups. Until now, for instance, it was unknown whether or
not these groups are residually finite. We will discuss the
answer to this and several other open questions after
providing the necessary background on surfaces of infinite
type. This work is joint with Nicholas G. Vlamis.

 Thursday December 01 at 34pm in Eck 308
 Jeremy Miller, Purdue
 Secondary representation stability and ordered configuration spaces

Abstract: I will describe a notion called secondary
representation stability. In joint work with Jennifer
Wilson, I proved that this phenomenon is present in the
homology of ordered configuration spaces of noncompact
surfaces. While representation stability in this context can
be formulated in terms of stabilizing by bringing a particle
in from infinity, secondary representation stability
involves bringing in two particles which orbit each other.
In particular, this stability pattern involves comparing
homology groups in different homological degrees. This
project can be thought of as a representation theoretic
analogue of secondary homological stability in the sense of
GalatiusKupersRandalWilliams.
Due to the high number of requests, we are no longer accepting speakers via selfinvitations.
For questions, contact