Manhattan Curves for Hyperbolic Surfaces with Cusps

TL;DR

This paper investigates the Manhattan curve linked to boundary-preserving Fuchsian representations of non-compact surfaces with cusps, proving its analyticity and establishing new rigidity results using Thermodynamic Formalism.

Contribution

It introduces the analyticity of the Manhattan curve for surfaces with cusps and extends existing rigidity results to this broader setting.

Findings

Proves the analyticity of the Manhattan curve for cusped surfaces.

Derives dynamical and geometric rigidity results for these representations.

Generalizes previous results by Burger and Sharp to non-compact cases.

Abstract

In this paper, we study an interesting curve, so-called the Manhattan curve, associated with a pair of boundary-preserving Fuchsian representations of a (non-compact) surface, especially representations corresponding to Riemann surfaces with cusps. Using Thermodynamic Formalism (for countable Markov shifts), we prove the analyticity of the Manhattan curve. Moreover, we derive several dynamical and geometric rigidity results, which generalize results of Marc Burger and Richard Sharp for convex-cocompact Fuchsian representations.