Junior Geometry and Topology in the Midwest

Oct. 13, 2018

University of Wisconsin, Madison

Consider a closed orientable surface of negative Euler characteristic. In joint work with A. Katok, we showed the flexibility of metric and topological entropies of geodesic flow in the class of negatively curved metrics of fixed total area. In this talk, we will discuss the aforementioned result and the flexibility of entropies under the additional restriction that the metrics we consider are conformally equivalent to a fixed hyperbolic metric. It turns out that some restrictions arise in conformal classes. Also, we will point out some geometric consequences for those families of Riemannian metrics (joint with T. Barthelm\'e).

Asilya Suleymanova (Indiana University)

**On the spectral geometry of manifolds with conic singularities**

In 1966 Mark Kac asked in his paper "Can one hear the shape of a drum?". Mathematically the question is formulated as follows. A drum is a domain in Euclidean space that is held along its boundary. When we play a drum we hear an infinite sequence of frequencies. The question is if one can determine the geometry of a domain using only information on this sequence. Nowadays the question is generalised to Riemannian manifolds and even to spaces with singularities. In this talk, we consider a manifold with conic singularities. We derive a detailed asymptotic expansion of the heat trace, then we investigate how the terms in the expansion reflect the geometry of the manifold. Can one hear a singularity?

Burns Healy (UW Milwaukee)

**Rigidity properties for hyperbolic generalizations**

Two commonly studied generalizations of Gromov hyperbolic groups are relatively hyperbolic and acylindrically hyperbolic groups. We'll begin by defining these properties and giving some examples. While in the geometric action setting, we get a well-defined boundary and quasi-isometry type for groups, this does not directly extend to these generalizations. In particular, we demonstrate the lack of a well defined limit set for acylindrical actions on hyperbolic spaces, even under the assumption of universality. We also prove a statement about relatively hyperbolic groups inspired by a remark asserted by Groves, Manning, and Sisto about the quasi-isometry type of combinatorial cusps.

Mark Pengitore (Ohio State)

**Nilpotent translation-like actions on real, complex, quaternionic, and Cayley hyperbolic spaces.**

The Gersten conjecture says that a group being hyperbolic is equivalent to having no Baumslag-Solitar subgroups. Even in the context of finitely presented groups, this is known to be false due to work of Brady. While there are weaker versions still open, we are interested in a geometric reformulation of the Gersten conjecture using translation-like actions. To be more specific, the geometric Gersten conjecture asks whether hyperbolicity is equivalent to having no translation-like actions by any Baumslag-solitar group. In this talk, we show that cocompact lattices in real rank 1 simple Lie groups admit translation-like actions by cocompact lattices in the unipotent part of the Iwasawa decomposition of the original Lie group. In particular, we demonstrate that Z^n acts translation-like on the fundamental group of any closed (n+1)-dimensional hyperbolic manifold. When n is greater than or equal to 2, we have an infinite collection of counterexamples to the geometric Gersten conjecture. This is work in progress with David Bruce Cohen and Ben McReynolds.

Kasia Jankiewicz (University of Chicago)

**Cubical dimension of C'(1/6) groups**

The cubical dimension of a group G is the infimum n such that G acts properly on an n dimensional CAT(0) cube complex. In my talk I will discuss small cancellation groups and a construction of C’(1/6) groups with cubical dimension greater than n.