Infima of length functions and dual cube complexes

TL;DR

This paper establishes lower bounds for length functions of curves on Teichmüller space using dual cube complexes, providing new estimates for curves with self-intersections and extending prior geometric bounds.

Contribution

It introduces a novel approach linking length functions to dual cube complexes, offering new bounds and estimates in Teichmüller theory.

Findings

Lower bounds for length functions depending on dual cube complexes

Estimates for the longest curves with k self-intersections

Extension of Basmajian's work on curve length estimates

Abstract

In the presence of certain topological conditions, we provide lower bounds for the infimum of the length function associated to a collection of curves on Teichm\"uller space that depend on the dual cube complex associated to the collection, a concept due to Sageev. As an application of our bounds, we obtain estimates for the `longest' curve with $k$ self-intersections, complementing work of Basmajian \cite{basmajian}.