Twisted Patterson-Sullivan measures and applications to amenability and coverings

Rémi Coulon (1) , Rhiannon Dougall (2) , Barbara Schapira (1) , Samuel Tapie (3)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 University of Bristol [Bristol]
3 LMJL - Laboratoire de Mathématiques Jean Leray

Let $\Gamma'<\Gamma$ be two discrete groups acting properly by isometries on a Gromov-hyperbolic space $X$. We prove that their critical exponents coincide if and only if $\Gamma'$ is co-amenable in $\Gamma$, under the assumption that the action of $\Gamma$ on $X$ is strongly positively recurrent, i.e. has a growth gap at infinity. This generalizes all previously known results on this question, which required either $X$ to be the real hyperbolic space and $\Gamma$ geometrically finite, or $X$ Gromov hyperbolic and $\Gamma$ cocompact. This result is optimal: we provide several counterexamples when the action is not strongly positively recurrent.

Mots clés: amenability; unitary representations; critical exponents; Twisted Patterson-Sullivan measure; strongly positively recurrent actions; growth gap at infinity; Gromov hyperbolic spaces

Domaines: Théorie des groupes [math.GR]; Systèmes dynamiques [math.DS]; Géométrie différentielle [math.DG]

Rémi Coulon, Rhiannon Dougall, Barbara Schapira, Samuel Tapie. Twisted Patterson-Sullivan measures and applications to amenability and coverings. 2019. ⟨hal-01881897v3⟩