Positively ratioed representations

Abstract

Let $S$ be a closed orientable surface of genus at least 2 and let $G$ be a semisimple real algebraic group of non-compact type. We consider a class of representations from the fundamental group of $S$ to $G$ called positively ratioed representations. These are Anosov representations with the additional condition that certain associated cross ratios satisfy a positivity property. Examples of such representations include Hitchin representations and maximal representations. Using geodesic currents, we show that the corresponding length functions for these positively ratioed representations are well-behaved. In particular, we prove a systolic inequality that holds for all such positively ratioed representations.