Anosov triangle reflection groups in SL(3,R)

TL;DR

This paper classifies all Anosov representations of compact hyperbolic triangle reflection groups into SL(3,R), revealing conditions for their existence and properties of their boundary maps, with implications for higher rank Lie group representations.

Contribution

It provides a complete classification of Anosov representations for these groups, identifying the Hitchin and Barbot components and their distinct boundary map characteristics.

Findings

Anosov representations are in the Hitchin component or Barbot component.

In the Barbot component, the product of generators has distinct real eigenvalues.

Boundary maps in the Barbot component are non-convex.

Abstract

We identify all Anosov representations of compact hyperbolic triangle reflection groups into the higher rank Lie group $\mathrm{SL}(3,\mathbb R)$. Specifically, we prove that such a representation is Anosov if and only if either it lies in the Hitchin component of the representation space, or it lies in the "Barbot component" and the product of the three generators of the triangle group has distinct real eigenvalues. Unlike representations in the Hitchin component, Anosov representations in the Barbot component have non-convex boundary maps.