Lyapunov exponents and stability properties of higher rank representations

TL;DR

This paper introduces the concept of proximal stability for holomorphic families of group representations and demonstrates its equivalence to the pluriharmonicity of associated Lyapunov exponents, extending previous results in the field.

Contribution

It generalizes prior work by defining proximal stability and proving its equivalence to Lyapunov exponent pluriharmonicity for higher rank representations.

Findings

Proximal stability is equivalent to Lyapunov exponent pluriharmonicity.

The notion extends the results of Deroin-Dujardin to higher rank representations.

Provides a new framework for analyzing stability in complex representations.

Abstract

Generalizing results of Deroin-Dujardin, we introduce the notion of proximal stability for a holomorphic family $ (\rho_{\lambda}) $ of representations $ \Gamma \to \text{SL}(d, \mathbb{C}) $, where $ \Gamma $ is a finitely generated group, and show that it is equivalent to the pluriharmonicity of Lyapunov exponents of the family (defined using random walks).