Organizers

Nathan Broaddus
Ian Leary

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OSU Topology Seminar
OSU Math Department

Participating Faculty

Dan Boros
Dan Burghelea
Indira Chatterji
Mike Davis
Tomasz Elsner
Jim Fowler
Honglin Min
Guido Mislin
Thomas Kerler

Previous Years

2015-2016
2014-2015
2013-2014
2012-2013
2011-2012
2010-2011

Autumn 2010
Thursday, October 21, 2010 at 3:30pm in CH 240

Fanny Kassel, University of Chicago

Compact quotients of rank-one groups

Abstract

A very natural question consists in describing all the compact manifolds that are locally modelled on some given homogeneous space G'/G, notably those that are quotients of G'/G by discrete groups acting properly discontinuously. I will consider the case when the homogeneous space is a rank-one semisimple group G endowed with the action of G×G by left and right multiplication. For G=SO(1,n), the question is related to the study of equivariant Lipschitz maps in hyperbolic spaces. I will also address similar problems in the setting of p-adic groups.

Thursday, November 4, 2010 at 2:30pm in CH 240

Alexander Borisov, University of Pittsburgh

Dynamical properties of polynomial automorphisms over finite and local fields and residual properties of mapping tori of free groups

Abstract

To every injective endomorphism of a free group Fk and any algebraic group (or group scheme) G, one can naturally associate an algebraic self-map of some Cartesian power of G. By a simple trick, due to Mark Sapir, periodic orbits of this map produce finite index subgroups in the corresponding mapping torus of Fk. We will discuss this connection and the relevant algebraic geometry results that imply that every mapping torus of a free group is virtually residually (finite p)-group for all but finitely many primes p. This is joint work with Mark Sapir.

Thursday, November 11, 2010 at 2:30pm in CH 240

Daniel Farley, Miami University

Finiteness properties of generalized Thompson groups

Abstract

Let T be a locally finite rooted simplicial tree. The space of ends of T, which we denote X, is a compact ultrametric space. A finite similarity structure S(X) assigns a finite (possibly empty) set S(B1,B2) of surjective similarities j: B1 → B2 to each pair of balls B1, B2 ⊂ X. The union of all sets S(B1,B2) (for varying B1 and B2) is also assumed to satisfy certain groupoid-like properties.

We are interested in the groups determined by finite similarity structures. Given S(X) (as above), we let G(S) denote the group of all self-homeomorphisms of X that are locally determined by S(X). That is, if h ∈ G(S), then, for every x ∈ X, there are balls B1 and B2, and a j ∈ S(B1,B2), such that x ∈ B1 and j|B1=h|B1.

For example, if we let T be the ordered infinite rooted binary tree, X be its space of ends, and, for any balls B1 and B2, S(B1,B2) be the singleton set containing the unique order-preserving similarity from B1 to B2, then G(S) is Thompson's group V. It appears that there are many other examples as well.

I will discuss joint work in progress with Bruce Hughes, in which we argue that a fairly general class of the groups G(S) have type F, i.e., have a classifying complex with finitely many cells in each dimension.

Thursday, November 18, 2010 at 2:30pm in CH 240

Jack Calcut, Oberlin College

The Torelli group, Donaldson's Theorem, and the Casson invariant in Artin Presentation Theory

Abstract

Artin presentations are discrete equivalents of planar open book decompositions of closed oriented 3-manifolds. An Artin presentation also determines a canonical smooth 4-manifold bounding the 3-dimensional open book. We will discuss several aspects of this discrete theory of smooth 4-manifolds, 3-manifolds, and links therein.

Thursday, December 2, 2010 at 2:30pm in CH 240

Spencer Dowdall, University of Chicago

Dilatation vs self-intersection number for point-pushing pseudo-Anosovs

Abstract

This talk is about the dilatations of pseudo-Anosov mapping classes obtained by pushing a marked point around a filling curve. After reviewing this "point-pushing" construction, I will give both upper and lower bounds on the dilatation in terms of the self-intersection number of the filling curve. I'll also give bounds on the least dilatation of any pseudo-Anosov in the point-pushing subgroup and describe the asymptotic dependence on self-intersection number. All of the upper bounds involve analyzing explicit examples using train tracks, and the lower bound is obtained by lifting to the universal cover and studying the images of simple closed curves.

Winter 2011
Thursday, January 20, 2011 at 2:30pm

Lucas Sabalka, Binghamton University, SUNY

On the Geometry of a Proposed Curve Complex Analogue for Out(Fn)

Abstract

The group Out(Fn) of outer automorphisms of the free group has been an object of active study for many years, yet its geometry is not well understood. Recently, effort has been focused on finding a hyperbolic complex on which Out(Fn) acts, in analogy with the curve complex for the mapping class group. In this talk on joint research with Dima Savchuk, we'll discuss some results about the geometry of edge splitting graph, or equivalently the separating sphere graph, a space on which Out(Fn) acts and a proposed candidate for a curve complex analogue. In particular, we have found quasi-flats of arbitrary dimension in this graph, showing it is not hyperbolic and has infinite asymptotic dimension.

Spring 2011
Thursday, April 7, 2011 at 2:30pm

Christophe Pittet, CMI University of Provence

A relation between the large scale geometry of a Lie group and the boundedness of its characteristic classes

Abstract

In 1982 M. Gromov proved that if G is a real linear algebraic group then all its characteristic classes are bounded. A weaker hypothesis on G implying the same conclusion is that the radical of G is linear. Searching for an even weaker hypothesis, we are lead to consider bounded 2-cocycles on G with all measurable representative unbounded. There is a clear geometric reformulation for the existence of such cocycles on G: the fundamental group of G is distorted in the universal cover of G although each cyclic subgroup of the fundamental group is undistorted.

This is work in progress with I. Chatterji, Y. De Cornulier (CNRS), and G. Mislin from OSU. A first part of the work, involving also L. Saloff-Coste from Cornell, is available at

http://www.springerlink.com/content/j05j2m6113762181/

Thursday, April 14, 2011 at 3pm (special time)

François Labourie, Orsay University

Kahn-Markovic work on surface subgroups in 3-manifolds

Thursday, April 21, 2011 at 2:30pm

Joel Louwsma, Caltech

Immersed surfaces in the modular orbifold

Abstract

A hyperbolic conjugacy class in the modular group PSL(2,Z) corresponds to a closed geodesic in the modular orbifold. Some of these geodesics virtually bound immersed surfaces, and some do not; the distinction is related to the polyhedral structure in the unit ball of the stable commutator length norm. We prove the following stability theorem: for every hyperbolic element of the modular group, the product of this element with a sufficiently large power of a parabolic element is represented by a geodesic that virtually bounds an immersed surface. This is joint work with Danny Calegari.

This seminar is supported by the National Science Foundation (NSF) under Grant No. and 1007059, and by the OSU Mathematics Research Institute (MRI).