Simple length rigidity for Kleinian surface groups and applications

TL;DR

This paper proves that Kleinian surface groups are uniquely determined by their simple marked length spectrum and explores applications in 3-manifold representations and symmetries of quasifuchsian space.

Contribution

It establishes simple length spectrum rigidity for Kleinian surface groups and applies this to 3-manifold representations and symmetries in quasifuchsian space.

Findings

Kleinian surface groups are determined by simple length spectrum

Discrete faithful representations are determined by boundary curve lengths

Diffeomorphisms preserving renormalized intersection are generated by mapping class group and conjugation

Abstract

We prove that a Kleinian surface groups is determined, up to conjugacy in the isometry group of $\mathbb H^3$, by its simple marked length spectrum. As a first application, we show that a discrete faithful representation of the fundamental group of a compact, acylindrical, hyperbolizable 3-manifold $M$ is similarly determined by the translation lengths of images of elements of $\pi_1(M)$ represented by simple curves on the boundary of $M$. As a second application, we show the group of diffeomorphisms of quasifuchsian space which preserve the renormalized intersection number is generated by the (extended) mapping class group and complex conjugation.