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 ${\rm Hom}(\Gamma,G)$. 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 ${\rm Hom}(\Gamma,G)$.