Geometric components of representation spaces via robust families of submanifolds

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 $\mathrm{Hom}(\Gamma ,G)$. 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 Teichmüller components for $G=\mathrm{SO}(p,q+1),\mathrm{SL}(p+1,\mathbb{R})$. Being based on very general principles, our approach might be suited for finding geometric components in various $\mathrm{Hom}(\Gamma ,G)$.

16 pages. Comments are welcome!