Kleinian viewpoints on higher rank worlds

TL;DR

This work surveys the program of translating Kleinian-group techniques to discrete subgroups of higher-rank Lie groups, with a focus on Anosov representations and their geometric/dynamic properties. It develops a robust linear-algebraic framework based on $\mathfrak a$, $J$, $\kappa$, flag varieties, and the Iwasawa cocycle to study growth, limit sets, and transversality, and then explores hyperconvex variants, combination theorems, and affine-action constructions. Key contributions include structural results for $(1,1,2)$-hyperconvex groups, several amalgamation and amalgam-type combination theorems for Anosov subgroups, and broad classes of proper affine actions for groups beyond the classical Kleinian setting. The results connect higher-rank discrete subgroups to convex-projective geometry, complex hyperbolic geometry, and semi-simple Lie-group dynamics, expanding the toolkit for understanding rigidity, limits, and deformations in higher rank.

Abstract

This paper is designed to attract people who work on real hyperbolic manifolds to consider thinking about discrete subgroups of higher rank Lie groups. To that end, we breezily discuss some applications of the ideas from the theory of Kleinian groups in the higher rank setting.

Theorems & Definitions (17)

• Theorem 3.1
• Theorem 3.2
• Theorem 4.1
• Theorem 4.2
• Theorem 4.3
• Theorem 4.4
• Theorem 4.5
• Theorem 5.1
• Corollary 5.2
• Theorem 5.3