We study locally compact, locally geodesically complete, locally CAT(κ) spaces (GCBA κ -spaces). We prove a Croke-type local volume estimate only depending on the dimension of these spaces. We show that a local doubling condition, with respect to the natural measure, implies pure-dimensionality. Then, we consider GCBA κ -spaces satisfying a uniform packing condition at some fixed scale r0 or a doubling condition at arbitrarily small scale, and prove several compactness results with respect to pointed Gromov-Hausdorff convergence. Finally, as a particular case, we study convergence and stability of M κ -complexes with bounded geometry.

We show the equivalences of several notions of entropy, such as a version of the topological entropy of the geodesic flow and the Minkowski dimension of the boundary, in metric spaces with convex geodesic bicombings satisfying a uniform packing condition. Similar estimates will be given in case of closed subsets of the boundary of Gromovhyperbolic metric spaces with convex geodesic bicombings. A uniform Ahlfors regularity of the limit set of quasiconvex-cocompact actions on Gromov-hyperbolic packed metric spaces with convex geodesic bicombing will be shown, implying a uniform rate of convergence to the entropy. As a consequence we prove the continuity of the critical exponent for quasiconvex-cocompact groups with bounded codiameter.

We show continuity under equivariant Gromov–Hausdorff convergence of the critical exponent of discrete, non-elementary, torsion-free, quasiconvex-cocompact groups with uniformly bounded codiameter acting on uniformly Gromov-hyperbolic metric spaces.

We study locally compact, locally geodesically complete, locally CAT(κ) spaces (GCBA κspaces). We prove a Croke-type local volume estimate only depending on the dimension of these spaces. We show that a local doubling condition, with respect to the natural measure, implies pure-dimensionality. Then we consider GCBA κ -spaces satisfying a uniform packing condition at some fixed scale r 0 or a doubling condition at arbitrarily small scale, and prove several compactness results with respect to pointed Gromov-Hausdorff convergence. Finally, as a particular case, we study convergence and stability of M κ -complexes with bounded geometry.

We show continuity of the critical exponent for quasiconvex-cocompact groups with bounded codiameter acting on packed, non-positively curved, Gromov-hyperbolic metric spaces.