On boundary maps of Anosov representations of a surface group to SL(3,R)

TL;DR

This paper investigates the uniqueness of boundary maps in Anosov representations of surface groups into SL(3,R), establishing conditions for their determination and analyzing the structure of the representation space.

Contribution

It characterizes when boundary maps uniquely determine Anosov representations and provides bounds on the number of mapping class group orbits in the representation space.

Findings

Boundary maps uniquely determine Anosov representations unless they factor through a reducible representation.

Provides a lower bound on the number of mapping class group orbits of Anosov representation components.

Discusses representations not distinguished by boundary maps and their implications.

Abstract

We prove that Anosov representations from a surface group to SL(3,R) are uniquely determined by their boundary maps if and only if they do not factor over a completely reducible representation. Furthermore we discuss representations not distinguished by their boundary maps, and we give a lower bound for the number of mapping class orbits of components of the space of Anosov representations.