Geometric components of representation spaces via robust families of submanifolds

TL;DR

This paper introduces robust families of submanifolds in linear Lie groups to identify geometric subspaces in representation spaces, providing a unified proof for higher Teichmüller components and suggesting broader applicability.

Contribution

It presents a new framework of robust submanifolds for Lie groups that yields geometric subspaces in representation spaces, unifying existing results on higher Teichmüller components.

Findings

Unified proof of higher Teichmüller components existence

Introduction of robust families of submanifolds for Lie groups

Potential for discovering geometric components in various representation spaces

Abstract

We introduce robust families of submanifolds for a linear Lie group $G$. We show that they give rise to geometric subspaces of the representation space ${\rm Hom}(\Gamma,G)$. As an application, we give a unified short proof of results of Beyrer and Kassel and of Benoist and Koszul about the existence of higher Teichm\"uller components for $G={\rm SO}(p,q+1),{\rm SL}(p+1,\mathbb{R})$. Being based on very general principles, our approach might be suited for finding geometric components in various ${\rm Hom}(\Gamma,G)$.