Çağrı Sert

Growth indicator and translation cone for 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 facts 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,
this generalizes several 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.