Flexibility and degeneracy around a theorem of Thurston

TL;DR

This paper presents two highly flexible and degenerate constructions related to Thurston's theorem, demonstrating non-rigidity and structural properties in Teichmüller space and representation limits.

Contribution

It introduces new flexible, degenerate examples in Teichmüller space and representation theory, revealing non-rigidity and structural insights.

Findings

Existence of geodesic segments stable under Lipschitz noise

Construction of open sets with limit cones over finite-sided polyhedra

Demonstration of maximal degeneracy in geometric and representation structures

Abstract

We give two flexible and degenerate constructions related to a theorem of Thurston. First, we produce geodesic segments for Thurston's asymmetric metric on Teichm\"uller space $\mathcal{T}(S_g)$ that remain geodesics after adding arbitrary $\varepsilon$-Lipschitz noise to all but one Fenchel-Nielsen coordinate. Then, for all $2 < n \leq 3g-3$ we construct open sets in $\mathcal{T}(S_g)^n$ for which the limit cones of the corresponding representations in $\mathrm{PSL}_2(\mathbb{R})^n$ are cones over explicit finite-sided polyhedra. Each construction is as degenerate as possible and has applications to the basic structure and local non-rigidity of the involved objects.