Isospectral finiteness on convex cocompact hyperbolic 3-manifolds

TL;DR

This paper proves that for a fixed set of geodesic lengths, only finitely many convex cocompact hyperbolic 3-manifolds exist with that spectrum, given certain topological constraints.

Contribution

It establishes the finiteness of convex cocompact hyperbolic 3-manifolds with a prescribed length spectrum under specific topological conditions.

Findings

Finiteness of manifolds with given length spectrum

Applicable to manifolds homotopy equivalent to a fixed 3-manifold

Excludes handlebody factors

Abstract

In this paper we show that a given set of lengths of closed geodesics, there are only finitely many convex cocompact hyperbolic 3-manifolds with that specified length spectrum, homotopy equivalent to a given 3-manifold without a handlebody factor, up to orientation preserving isometries.