Balanced ideals and domains of discontinuity of Anosov representations

TL;DR

This paper studies the action of Anosov subgroups on flag manifolds, showing that all cocompact domains of discontinuity are derived from a known construction, and introduces a new compactification for certain locally symmetric spaces.

Contribution

It proves that all cocompact domains of discontinuity for $ riangle$-Anosov representations are from a systematic construction, with some exceptions, and introduces a new compactification for specific maximal representations.

Findings

All cocompact domains of discontinuity are from the known construction, except in low rank cases.

Determined which flag manifolds admit these domains and counted them in some cases.

Introduced a new compactification for locally symmetric spaces from maximal representations into $ ext{Sp}(4n+2, r)$.

Abstract

We consider the action of Anosov subgroups of a semi-simple Lie group on the associated flag manifolds. A systematic approach to construct cocompact domains of discontinuity for this action was given by Kapovich, Leeb and Porti in arXiv:1306.3837. For $\Delta$-Anosov representations, we prove that every cocompact domain of discontinuity arises from this construction, up to a few exceptions in low rank. Then we compute which flag manifolds admit these domains and, in some cases, the number of domains. We also find a new compactification for locally symmetric spaces arising from maximal representations into $\mathrm{Sp}(4n+2, \mathbb{R})$.