Which reducible representations are Anosov?

TL;DR

This paper characterizes when reducible representations are Anosov by examining eigenvalue magnitudes of their block components, revealing that many hyperbolic groups' character varieties lack reducible Anosov representations.

Contribution

It provides a new eigenvalue-based criterion for identifying Anosov reducible representations, extending understanding of their structure within character varieties.

Findings

Connected components of character varieties of many hyperbolic groups contain only irreducible Anosov representations.

The eigenvalue magnitude condition characterizes the Anosov property for reducible representations.

Many non-elementary hyperbolic groups have character variety components devoid of reducible Anosov representations.

Abstract

We give a characterization of the Anosov condition for reducible representations in terms of the eigenvalue magnitudes of the irreducible block factors of its block diagonalization. As in previous work, these Anosov representations comprise a collection of bounded convex domains in a finite-dimensional vector space, and this perspective allows us to conclude for many non-elementary hyperbolic groups that connected components of the character variety which consist entirely of Anosov representations do not contain reducible representations.