Thurston's metric on Teichm\"uller space of semi-translation surfaces

TL;DR

This paper introduces new symmetric and asymmetric pseudo-metrics on the Teichmüller space of semi-translation surfaces, establishing their properties and exploring their relationships with 1-Lipschitz maps, with some conjectures remaining open.

Contribution

It defines and analyzes new symmetric and asymmetric Thurston-type metrics on the Teichmüller space of semi-translation surfaces, including their completeness and conjectural properties.

Findings

The symmetric pseudo-metrics are complete.

Asymmetric metrics are defined and their properties studied.

A conjecture relates the asymmetric metrics to 1-Lipschitz polygon maps.

Abstract

The present paper is composed of two parts. In the first one we define two pseudo-metrics $L_F$ and $K_F$ on the Teichmu\"uller space of semi-translation surfaces $\mathcal{TQ}_g(\underline k,\epsilon)$, which are the symmetric counterparts to the metrics defined by William Thurston on $\mathcal{T}_g^n$. We prove some nice properties of $L_F$ and $K_F$, most notably that they are complete pseudo-metrics. In the second part we define their asymmetric analogues $L_F^a$ and $K_F^a$ on $\mathcal{T Q}_g^{(1)}(k, \epsilon)$ and prove that their equality depends on two statements regarding 1-Lipschitz maps between polygons. We are able to prove the first statement, but the second one remains a conjecture: nonetheless, we explain why we believe it is true.