Growth indicator and translation cone for Gromov hyperbolic groups

We first introduce a class of metric-like functions on hyperbolic groups, called hyperbolic metric potentials. This is a class of functions general enough to include word-metrics, quasi-morphisms, and the fundamental weights of Anosov representations. Then, given a tuple (f1,...,fd) of such functions, we introduce the notion of translation cone, an analogue of the limit cone introduced in the setting of linear algebraic groups by Benoist in the 90s. We establish analogues of Benoist's results as well as additional hyperbolic features on this cone. We then turn to a more precise asymptotic analysis: counting. We introduce the analogue of the growth indicator function, introduced in early 2000s by Quint again in the linear setting. We show that this function is always strictly concave and C1, which generalize results of Quint, Sambarino, Kim-Oh-Wang, etc. Finally, we relate this function to a multi-dimensional generalization of the Manhattan curve, which is a curve of Poincaré exponents, introduced in dimension 2 by Burger in the 90s, and recently studied in our more general setup by Tanaka, and Cantrell--Tanaka. Joint work with Stephen Cantrell and Eduardo Reyes.