Finite height lamination spaces for surfaces

TL;DR

This paper introduces a new parameter space called  for describing finite height measured laminations on surfaces, and demonstrates how these laminations relate to actions of the surface's fundamental group on -trees.

Contribution

It develops a novel framework using  to parametrize finite height laminations and links these to -tree actions of the fundamental group.

Findings

 effectively parametrizes finite height laminations.

Every finite height lamination induces an -tree action of .

The framework generalizes existing lamination theories.

Abstract

We describe spaces of essential finite height (measured) laminations in a surface $S$ using a parameter space we call $\mathbb S$, an ordered semi-ring. We show that for every finite height essential lamination $L$ in $S$, there is an action of $\pi_1(S)$ on an $\mathbb S$-tree dual to the lift of $L$ to the universal cover of $S$.