A collar lemma for partially hyperconvex surface group representations

Abstract

We show that a collar lemma holds for Anosov representations of fundamental groups of surfaces into $\SL(n,\R)$ that satisfy partial hyperconvexity properties inspired from Labourie's work. This is the case for several open sets of Anosov representations not contained in higher rank Teichm\"uller spaces, as well as for $\Theta$-positive representations into $\SO(p,q)$ if $p\geq 4$. We moreover show that 'positivity properties' known for Hitchin representations, such as being positively ratioed and having positive eigenvalue ratios, also hold for partially hyperconvex representations.