Transverse measures to infinite type laminations

TL;DR

This paper characterizes the cone of transverse measures for geodesic laminations on infinite type hyperbolic surfaces, describing its structure as an inverse limit of finite-dimensional cones and simplices, and explores when it admits a base.

Contribution

It provides an explicit inverse limit description of the cone of transverse measures and characterizes when this cone admits a base, including the construction of exotic examples.

Findings

The cone of transverse measures can be described as an inverse limit of finite-dimensional cones.

Many laminations have a base that is a Choquet simplex, often infinite-dimensional.

Every Choquet simplex can be realized as a base for some lamination on an infinite type surface.

Abstract

We study the cone of transverse measures to a fixed geodesic lamination on an infinite type hyperbolic surface. Under simple hypotheses on the metric, we give an explicit description of this cone as an inverse limit of finite-dimensional cones. We study the problem of when the cone of transverse measures admits a base and show that such a base exists for many laminations. Moreover, the base is a (typically infinite-dimensional) simplex (called a Choquet simplex) and can be described explicitly as an inverse limit of finite-dimensional simplices. We show that on any fixed infinite type hyperbolic surface, every Choquet simplex arises as a base for some lamination. We use our inverse limit description and a new construction of geodesic laminations to give other explicit examples of cones with exotic properties.