###### 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 `F _{k}` 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

`F`. 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

_{k}`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(B _{1},B_{2})` of surjective
similarities

`j: B`to each pair of balls

_{1}→ B_{2}`B`. The union of all sets

_{1}, B_{2}⊂ X`S(B`(for varying

_{1},B_{2})`B`and

_{1}`B`) is also assumed to satisfy certain groupoid-like properties.

_{2}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
`B _{1}` and

`B`, and a

_{2}`j ∈ S(B`, such that

_{1},B_{2})`x ∈ B`
and

_{1}`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 `B _{1}` and

`B`,

_{2}`S(B`be the singleton set containing the unique order-preserving similarity from

_{1},B_{2})`B`to

_{1}`B`, then

_{2}`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.

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

### Lucas Sabalka, Binghamton University, SUNY

#### On the Geometry of a Proposed Curve Complex Analogue for `Out(F`_{n})

_{n})

##### Abstract

The group `Out(F _{n})` 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(F`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

_{n})`Out(F`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.

_{n})###### 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

###### 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.