Commentarii Mathematici Helvetici

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Volume 94, Issue 2, 2019, pp. 273–345
DOI: 10.4171/CMH/461

Published online: 2019-04-17

Positively ratioed representations

Giuseppe Martone[1] and Tengren Zhang[2]

(1) University of Michigan, Ann Arbor, USA
(2) National University of Singapore, Singapore

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.

Keywords: Hitchin representations, maximal representations, higher Teichm├╝ller theory, geodesic currents

Martone Giuseppe, Zhang Tengren: Positively ratioed representations. Comment. Math. Helv. 94 (2019), 273-345. doi: 10.4171/CMH/461