Entropy, Critical Exponent and Immersed Surfaces in Hyperbolic 3-Manifolds

TL;DR

This paper establishes bounds relating the topological entropy of geodesic flows on immersed surfaces in hyperbolic 3-manifolds to the critical exponent of their fundamental groups, revealing rigidity and geometric properties.

Contribution

It introduces new inequalities connecting entropy and critical exponent for immersed surfaces, and characterizes constants as geodesic stretches in certain cases.

Findings

Bounds on entropy in terms of critical exponent with constants ≤ 1

Exact characterization of constants as geodesic stretches when immersion is an embedding

Rigidity phenomena derived from the entropy-critical exponent inequality

Abstract

We consider a $\pi_{1}$--injective immersion $f:\Sigma\to M$ from a compact surface $\Sigma$ to a hyperbolic 3--manifold $M$. Let $\Gamma$ denote the copy of $\pi_{1}\Sigma$ in $\mathrm{Isom}({\mathbb{H}}^{3})$ induced by the immersion and $\delta(\Gamma)$ be the critical exponent. Suppose $\Gamma$ is convex cocompact and $\Sigma$ is negatively curved, we prove that there are two geometric constants $C_{1}(\Sigma,M)$ and $C_{2}(\Sigma,M)$ not bigger than $1$ such that $C_{1}(\Sigma,M)\cdot\delta_{\Gamma}\leq h(\Sigma)\leq C_{2}(\Sigma,M)\cdot\delta_{\Gamma}$, where $h(\Sigma)$ is the topological entropy of the geodesic flow on. When $f$ is an embedding, we show that $C_{1}(\Sigma,M)$ and $C_{2}(\Sigma,M)$ are exactly the geodesic stretches (a.k.a. Thurston's intersection number) with respect to certain Gibbs measures. Moreover, we prove the rigidity phenomenon arising from this inequality. Lastly, as an application, we discuss immersed minimal surfaces in hyperbolic 3--manifolds and these discussions lead us to results similar to A. Sanders' work on the moduli space of $\Sigma$ introduced by C. Taubes.