Cataclysms for Anosov representations

TL;DR

This paper introduces a new type of deformation called cataclysms for $ heta$-Anosov representations into semisimple Lie groups, generalizing known deformations in Teichmüller theory and higher rank settings.

Contribution

It constructs and analyzes cataclysm deformations for $ heta$-Anosov representations, extending Thurston's and Dreyer's work, and studies their properties and injectivity.

Findings

Cataclysm deformations are additive and compatible with group homomorphisms.

Deformation is injective for Hitchin representations.

Deformation is not injective in general for $ heta$-Anosov representations.

Abstract

In this paper, we construct cataclysm deformations for $\theta$-Anosov representations into a semisimple non-compact connected real Lie group $G$ with finite center, where $\theta \subset \Delta$ is a subset of the simple roots that is invariant under the opposition involution. These generalize Thurston's cataclysms on Teichm\"uller space and Dreyer's cataclysms for Borel-Anosov representations into $\mathrm{PSL}(n, \mathbb{R})$. We express the deformation also in terms of the boundary map. Furthermore, we show that cataclysm deformations are additive and behave well with respect to composing a representation with a group homomorphism. Finally, we show that the deformation is injective for Hitchin representations, but not in general for $\theta$-Anosov representations.