Entropy of embedded surfaces in quasi-fuchsian manifolds

TL;DR

This paper investigates the relationship between the critical exponent of quasi-Fuchsian groups acting on hyperbolic 3-space and on embedded invariant disks, establishing rigidity results for specific cases.

Contribution

It introduces new rigidity theorems for embedded surfaces in hyperbolic space under Fuchsian and quasi-Fuchsian group actions.

Findings

Rigidity theorem for Fuchsian group actions on embedded surfaces.

Rigidity theorem for negatively curved surfaces under quasi-Fuchsian actions.

Comparison of critical exponents for different group actions on hyperbolic space.

Abstract

We compare critical exponent for quasi-Fuchsian groups acting on the hyperbolic 3-space, $\mathbb{H}^3$, and on invariant disks embedded in $\mathbb{H}^3$. We give a rigidity theorem for all embedded surfaces when the action is Fuchsian and a rigidity theorem for negatively curved surfaces when the action is quasi-Fuchsian.