The space of measured foliations of the hexagon

TL;DR

This paper explores the space of measured foliations on a hexagon, revealing its piecewise-linear structure and its role as a boundary for hyperbolic structures with geodesic boundary and right angles.

Contribution

It characterizes the space of measured foliations on a hexagon and its natural cell-decomposition, connecting it to hyperbolic structures with boundary.

Findings

The space of measured foliations on a hexagon has a natural piecewise-linear structure.

This space acts as a boundary for hyperbolic structures with geodesic boundary and right angles.

The paper describes the geometric and combinatorial properties of these spaces.

Abstract

The theory of geometric structures on a surface with nonempty boundary can be developed by using a decomposition of such a surface into hexagons, in the same way as the theory of geometric structures on a surface without boundary is developed using the decomposition of such a surface into pairs of pants. The basic elements of the theory for surfaces with boundary include the study of measured foliations and of hyperbolic structures on hexagons. It turns out that there is an interesting space of measured foliations on a hexagon, which is equipped with a piecewise-linear structure (in fact, a natural cell-decomposition), and this space is a natural boundary for the space of hyperbolic structures with geodesic boundary and right angles on such a hexagon. In this paper, we describe these spaces and the related structures.