Properties of sets of isometries of Gromov hyperbolic spaces

Abstract: We prove an inequality concerning isometries of a Gromov hyperbolic metric space, which does not require the space to be proper or geodesic. It involves the joint stable length, a hyperbolic version of the joint spectral radius, and shows that sets of isometries behave like sets of $2 \times 2$ real matrices. Among the consequences of the inequality, we obtain the continuity of the joint stable length and an analogue of Berger-Wang theorem.