Junior Geometry and Topology in the Midwest

April 22, 2017

University of Wisconsin, Madison

Birman and Hilden ask: given finite branched cover X over the 2-sphere, does every homeomorphism of the sphere lift to a homeomorphism of X? For covers of degree 2, the answer is yes, but the answer is sometimes yes and sometimes no for higher degree covers. In joint work with Ghaswala, we completely answer the question for cyclic branched covers. When the answer is yes, there is an embedding of the mapping class group of the sphere into a finite quotient of the mapping class group of X. In a family where the answer is no, we find a presentation for the group of isotopy classes of homeomorphisms of the sphere that do lift, which is a finite index subgroup of the mapping class group of the sphere.

Harrison Bray (Michigan)

**A measure of maximal entropy for nonstrictly convex Hilbert geometries**

Strictly convex Hilbert geometries naturally generalize constant negatively curved Riemannian geometries, and the geodesic flow on quotients has been well-studied by Benoist, Crampon, Marquis, and others. In contrast, nonstrictly convex Hilbert geometries in three dimensions have the feel of nonpositive curvature, but also have a fascinating geometric irregularity which forces the geodesic flow to avoid direct application of existing nonuniformly hyperbolic theory. In this talk, we will introduce the audience to geometric techniques for studying the geodesic flow which can be adapted to this setting. The main result is construction of a measure of maximal entropy which is ergodic for the geodesic flow.

Shi Wang (Bloomington)

**Simplicial volume of nonpostitively curved manifolds**

In this talk, I will discuss the notion of simplicial volume introduced by Gromov and Thurston. For nonpositively curved manifolds, Gromov conjectured that negative Ricci curvature implies postivitity of simplicial volume. I will talk about some recent work joint with Chris Connell. We show that, under a stronger curvature condition, the simplicial volume is always positive, this answers a special case of Gromov's problem.

Juliette Bavard (UChicago)

**Around a big mapping class group**

The mapping class group of the plane minus a Cantor set arises naturally in many dynamical contexts. To study this ‘big mapping class group’, we can consider its action on a Gromov-hyperbolic graph called the 'ray graph'. In this talk, I will present this big mapping class group and its ray graph, and explain why the ray graph has infinite diameter and is Gromov-hyperbolic.

Mark Bell (UIUC)

**Computations with curves on surfaces**

A pair of curves on a surface can appear extremely complicated and so it can be difficult to compute properties such as their intersection number. We will discuss a new argument that, when the curve is given by its intersections with the edges of an ideal triangulation, there is always a "reduction" to a simpler configuration in which such calculations are straightforward. This relies on finding an edge flip or a (power of a) Dehn twist that decreases the complexity of a curve by a definite fraction.