Moon Duchin

Length spectra and degeneration of flat metrics

In this paper we consider flat metrics (semi-translation structures) on surfaces of finite type. There are two main results. The first is a complete description of when a set of simple closed curves is spectrally rigid, that is, when the length vector determines a metric among the class of flat metrics. Secondly, we give an embedding into the space of geodesic currents and use this to obtain a compactification for the space of flat metrics. The geometric interpretation is that flat metrics degenerate to mixed structures on the surface: part flat metric and part measured foliation.

Statistical hyperbolicity in groups

In this paper, we introduce a geometric statistic called the sprawl of a group with respect to a generating set, based on the average distance in the word metric between pairs of words of equal length. The sprawl quantifies a certain obstruction to hyperbolicity. Group presentations with maximum sprawl (ie without this obstruction) are called statistically hyperbolic. We first relate sprawl to curvature and show that nonelementary hyperbolic groups are statistically hyperbolic, then give some results for products and for certain solvable groups. In free abelian groups, the word metrics are asymptotic to norms induced by convex polytopes, causing several kinds of group invariants to reduce to problems in convex geometry. We present some calculations and conjectures concerning the values taken by the sprawl statistic for the group Z d as the generators vary, by studying the space R d with various norms.

The geometry of spheres in free abelian groups

We study word metrics on

Divergence of Geodesics in Teichmüller Space and the Mapping Class Group

{\mathbb{Z}^d}

Statistical Hyperbolicity in Teichmüller Space

by developing tools that are fine enough to measure dependence on the generating set. We obtain counting and distribution results for the words of length n. With this, we show that counting measure on spheres always converges to cone measure on a polyhedron (strongly, in an appropriate sense). Using the limit measure, we can reduce probabilistic questions about word metrics to problems in convex geometry of Euclidean space. We give several applications to the statistics of “size-like” functions.

Equations in nilpotent groups

What is the average distance between two points on the sphere of radius n in a normed space? There is a natural choice of measure making this average-distance statistic into an affine invariant, and we explore the conjecture that the ℓ2 and ℓ∞ norms provide the two extreme values of this invariant on for every d.