University of Michigan
Department of Mathematics
Fall 2009 Topology Seminar
Thursdays 3-4, 4096 East Hall

Schedule of Talks

Abstracts

Homotopy Theory and Spaces of Representations

Using spaces of homomorphisms and the descending central series of the free groups, simplicial spaces are constructed for each integer q>1 and every topological group G,
with realizations B(q,G) that filter the classifying space BG. In particular for q=2 this yields a single space B(2,G) assembled from all the n-tuples of commuting elements in G.
Homotopy properties of the B(q,G) will be described for finite groups, and cohomology calculations provided for compact Lie groups. Recent results on understanding both the
number and stable homotopy type of the components of related spaces of representations will also be discussed.

Superinjective Simplicial Maps of the Complexes of Curves on Nonorientable Surfaces

We prove that each superinjective simplicial map of the complex of curves of a compact, connected, nonorientable surface is induced by a homeomorphism of the
surface, if g + n is at most 3 or g + n is at least 5, where g is the genus of the surface and n is the number of the boundary components.

Periodic maximal flats are not peripheral

We prove that every finite volume locally symmetric space M contains a compact set K with the property that no periodic maximal flat can be homotoped so that it is disjoint of K. This
is joint work with Alexandra Pettet.

Rank gradient of cyclic covers

If M is a hyperbolic 3-manifold with a map from the fundamental group of M onto Z, the associated family of finite cyclic covers has non-positive rank gradient if and only if the kernel
of this map is finitely generated. I will describe a proof using actions on trees and, time permitting, one using carrier graphs.

High Distance Knots in any 3-Manifold

Let M be a closed 3-manifold with a given Heegaard splitting. We show that after a single stabilization, some core of the stabilized splitting has arbitrarily high distance with respect to the splitting surface. This generalizes a result of Minsky, Moriah, and Schleimer for knots in S^3. We also show that in the complex of curves, handlebody sets are either coarsely distinct or identical. We define the coarse mapping class group of a Heeegaard splitting, and show that if (S, V, W) is a Heegaard splitting of genus greater than or equal to 4, then CMCG(S,V,W) \cong MCG(S, V,W). This is joint work with Marion Moore.

JSJ splittings, deformation spaces, and the isomorphy problem for hyperbolic groups

I will explain a solution of the isomorphy problem for hyperbolic groups in presence of torsion. This involves computing some structural invariant of these groups, like JSJ deformation spaces, and deformation spaces of splittings over finite groups. But instead of computing the usual JSJ decomposition, we have to construct a JSJ decomposition for splittings which produce non-trivial Dehn twists. For instance, this kind of JSJ decomposition gives a non-trivial canonical decomposition of fundamental groups of 2-orbifolds with mirrors. This is a common work with Francois Dahmani.

Fully irreducible outer automorphisms of a free group

Fully irreducible outer automorphisms of a free group are analogous to loxodromic isometries of hyperbolic space, or to pseudo-Anosov elements of the mapping class group of a surface. We develop methods for constructing customized fully irreducible elements of a free group F of rank k. For example, there exists for any matrix A in GL(k,Z) a non-geometric fully irreducible element inducing the action of A on the non-abelian free group of rank k. This is an analogue of a well-known theorem for the mapping class group. This is joint work with Matt Clay.

Commensurators and arithmetic lattices in right-angled buildings

For G a locally compact topological group, recall that a discrete subgroup H is a lattice in G if H\G has finite volume. A lattice H is called a uniform lattice if H\G is actually compact.  In the Lie group setting, Margulis proved that a lattice is arithmetic if and only if its commensurator is dense in G.  We consider the case when G is the automorphism group of a locally finite polyhedral complex X.  If X is a tree, results of Liu, Bass-Kulkarni, and Leighton show that the commensurator of every uniform lattice is dense in G.  When X is a right-angled building, we construct new examples of lattices and use them to show that the commensurator of the "standard uniform lattice" is dense in G.This is joint work with Anne Thomas. 

Volume Distortion in Groups

Suppose H is a finitely presented group, and let w be a word representing the identity in H. Then w can be written as the product of conjugates of relators; the minimal such number is defined to be the area of w. If H is a subgroup of G, also finitely presented, then it is possible that the area of w in G is much less than its area in H. We will define the area distortion function, which provides a measure for this difference, and then generalize this to a volume distortion function. We will provide bounds in terms of the Dehn functions of the groups and find the volume distortion for abelian subgroups of abelian-by-cyclic groups.

Twisted spin bordism and twisted K-theory

Let X be a space equipped with a three dimensional integral cohomology class T. In this talk, we'll show how we can use T to define both the twisted complex K-theory of X and the twisted Spinc cobordism of X . Hopkins and Hovey proved that the (untwisted) complex K-theory of X is related to the Spin^c bordism of X via an isomorphism of Conner-Floyd type. We investigate the analogous question for the twisted theories. This investigation leads to a clarification formula for twisted K-theory.

For more information, contact Enrique Torres-Giese.

If you want to check the list of seminars held in previous terms, click on the appropriate link below.

Winter 2009, Fall 2008, Winter 2008, Fall 2007, Winter 2007, Fall 2006, Winter 2006, Fall 2005, Winter 2005, Fall 2004, Winter 2004, Fall 2003, Winter 2003, Fall 2002, the year 2000-1.

This page last updated on November 4, 2009.