Planes in degenerate 3-manifolds

TL;DR

This paper investigates the structure of totally geodesic planes in certain hyperbolic 3-manifolds, proving a rigidity phenomenon and finiteness results for such surfaces.

Contribution

It establishes a Ratner-type rigidity result for minimal invariant sets and proves finiteness of compact totally geodesic surfaces in specific infinite volume hyperbolic 3-manifolds.

Findings

A closed minimal PSL(2,R)-invariant subset is either a totally geodesic surface or the entire manifold.

Finiteness of compact totally geodesic surfaces in infinite volume hyperbolic 3-manifolds without parabolics.

The results extend understanding of geometric structures in degenerate hyperbolic 3-manifolds.

Abstract

We study totally geodesic planes in hyperbolic 3-manifolds $M$ having incompressible core and degenerate ends. We prove a Ratner-type phenomenon: a closed minimal $PSL(2,R)-$invariant subset of $M$ is either an immersed totally geodesic surface or all of $M$. We also show that for an arbitrary infinite volume hyperbolic 3-manifold $M$ without parabolics and with finitely generated fundamental group, the number of compact totally geodesic surfaces in $M$ is finite.