Hyperconvex representations and exponential growth

TL;DR

This paper investigates the properties of limit cones and growth indicators for certain representations of fundamental groups of negatively curved manifolds into semi-simple Lie groups, analyzing their variation with the representation.

Contribution

It introduces a study of the limit cone and growth indicator function for a class of representations with boundary maps, extending understanding of their behavior in geometric group theory.

Findings

Analysis of the limit cone and growth indicator for these representations.

Results on how these objects vary with the representation.

Connections to the geometry of negatively curved manifolds.

Abstract

Let $G$ be a real algebraic semi-simple Lie group and $\Gamma$ be the fundamental group of a compact negatively curved manifold. In this article we study the limit cone, introduced by Benoist, and the growth indicator function, introduced by Quint, for a class of representations $\rho:\Gamma\to G$ admitting a equivariant map from $\partial\Gamma$ to the Furstenberg boundary of $G$'s symmetric space together with a transversality condition. We then study how these objects vary with the representation.