Finite step Rigidity of Hitchin representations and special Margulis-Smilga spacetimes

TL;DR

This paper establishes finite step rigidity results for certain spectral invariants of Hitchin representations and Margulis-Smilga spacetimes, using the escape from subvarieties lemma, and extends similar results to Cartan spectra of representations in semisimple Lie groups.

Contribution

It introduces finite step rigidity results for spectral invariants of Hitchin representations, Margulis-Smilga spacetimes, and Cartan spectra, expanding the understanding of spectral rigidity in these contexts.

Findings

Finite step rigidity for Jordan-Lyapunov spectra of Hitchin representations.

Finite step rigidity for Margulis-Smilga invariant spectra of specific spacetimes.

Finite step rigidity for Cartan spectra of representations in semisimple Lie groups.

Abstract

In this article we use the "escape from subvarieties lemma" introduced by Eskin--Mozes--Oh to prove finite step rigidity results for the Jordan-Lyapunov projection spectra of Hitchin representations and the Margulis-Smilga invariant spectra of some special Margulis-Smilga spacetimes. In the process, we also prove a similar finite step rigidity result for the Cartan spectra of representations of a finitely generated group inside a connected semisimple real algebraic Lie group with trivial center.