Bending, entropy and proper affine actions of surface groups

TL;DR

The paper investigates the relationship between surface group representations near the Fuchsian locus, proper affine actions, and entropy, revealing new neighborhoods where these properties hold and characterizing critical points of entropy.

Contribution

It provides explicit neighborhoods around the Fuchsian locus in quasifuchsian space where non-Fuchsian representations induce proper affine actions and characterizes entropy critical points.

Findings

Existence of explicit neighborhoods with proper affine actions for non-Fuchsian representations.

Larger neighborhoods where entropy critical points are confined to the Fuchsian locus.

Identification of conditions linking entropy, bending, and affine actions.

Abstract

We show that for any closed surface $S$ there is an explict neighborhood $V$ of the fuchsian locus in quasifuchsian space $\mathsf{QF}(S)$ such that for every representation $\rho\in V$ which is not fuchsian, there is a proper affine action on $\mathfrak{sl}(2,\mathbb{C})$ with linear part $\mathsf{Ad}(\rho)$. We further show that there is a larger neighborhood $U$ of the Fuchsian locus so that every critical point of the entropy function in $U$ lies on the Fuchsian locus.