Teichmuller space and geodesic currents

Introduction to geodesic currents

Introduction to measured laminations and geodesic currents, as a first introductory lecture for the session on geodesic currents and Teichmueller theory.

A qualitative description of the horoboundary of the Teichmüller metric 

In this intro talk for the session, we describe what moduli and Teichmüller spaces are and examine the space of flat tori in detail. We also discuss hyperbolic structures on higher genus surfaces, ending with a description of certain nice coordinates for their Teichmüller spaces.

A qualitative description of the horoboundary of the Teichmüller metric 

The horofunction compactification of a metric space keeps track of the possible limits of balls whose centers go off to infinity. This construction was introduced by Gromov, and although it is usually hard to visualize, it has proved to be a useful tool for studying negatively curved spaces. In this talk I will explain how, under some metric assumptions, the horofunction compactification is a refinement of the significantly simpler visual compactification. I will then go over how this relation allows us to use the simplicity of the visual compactification to get geometric and topological properties of the horofunction compactification. Most of these applications will be in the context of Teichmüller spaces with respect to the Teichmüller metric, where the relation allows us to prove, among other things, that Busemann points are not dense within the horoboundary and that the horoboundary is path connected.

Paper: A qualitative description of the horoboundary of the Teichmüller metric 

Stars at infinity in Teichmüller Space

In 2005, Anders Karlsson introduced stars at infinity, a construction which provides added structure to any compactification of a metric space. The stars in the boundary, defined using just the metric itself, are powerful enough to give new proofs to well-known results across geometry and dynamical systems. During the talk, I will define stars, motivate their importance, and state a new result on stars in the Thurston boundary of Teichmüller space. This is joint work with Moon Duchin.

Paper: Stars at infinity in Teichmüller space, with Moon Duchin

Earthquakes on the once-punctured torus

We study earthquake deformations on Teichmüller space associated with simple closed curves of the once-punctured torus. We describe two methods to get an explicit form of the earthquake deformation for any simple closed curve. The first method is rooted in hyperbolic geometry, the second representation theory. The two methods align, providing both a geometric and an algebraic interpretation of the earthquake deformations. Two families of curves are used as examples. Examining the limiting behaviour of each gives insight into earthquakes about measured geodesic laminations, of which simple closed curves are a special case.

Paper: Earthquakes on the once-punctured torus

Non-positive curvature, biautomaticity, and the space of currents

In these talks we will investigate the relationship between groups acting geometrically on spaces with various notions of non-positive curvature and biautomaticity. The latter property implies strong algorithmic properties for a group. We will resolve the question of whether every hierarchically hyperbolic group is biautomatic and explain how the solution uses the space of currents. Joint work with Motiejus Valiunas.

Paper: Commensurating HNN-extensions: hierarchical hyperbolicity and biautomaticity, with Motiejus Valiunas

Superdensity and bounded geodesics in moduli space

Superdensity is a quantitative density condition on a flow. In this talk, we will show that the linear flow on a translation surface is superdense only if the associated Teichmüller geodesic is bounded. The result is motivated by and generalizes earlier work of Beck-Chen, where superdensity is first defined, and is reminiscent of Masur's Criterion. 

Equidistribution of the intersection points on hyperbolic surfaces

In this talk, we will use the theory of geodesic currents to show the intersection points between closed geodesics on a finite volume hyperbolic surface are equidistributed.