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 get a boundary 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.Comment: 36 page

Rational growth in the Heisenberg group

A group presentation is said to have rational growth if the generating series associated to its growth function represents a rational function. A long-standing open question asks whether the Heisenberg group has rational growth for all finite generating sets, and we settle this question affirmatively. We also establish almost-convexity for all finite generating sets. Previously, both of these properties were known to hold for hyperbolic groups and virtually abelian groups, and there were no further examples in either case. Our main method is a close description of the relationship between word metrics and associated Carnot-Caratheodory Finsler metrics on the ambient Lie group. We provide (non-regular) languages in any word metric that suffice to represent all group elements.Comment: Version 2 contains bug fixes and an added application to almost-convexit