arXiv subjects
Publications and source records attributed to Nicola Cavallucci.
We study groups of isometries of packed, geodesically complete, CAT$(0)$-spaces for which the systole at every point is smaller than a universal constant depending only on the packing, deducing strong rigidity results. We show that if a space as above has some negative curvature behaviour then it cannot support a thin action: this generalizes the classical Margulis Lemma to a broader class of spaces.
We reprove in an easier way a result of Brian Clarke: the completion of the space of Riemannian metrics of a compact, orientable smooth manifold with respect to the $L^2$-distance is CAT$(0)$. In particular we show that this completion is isometric to the space of $L^2$-maps from a standard probability space to a fixed CAT$(0)$ space.
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 show the equivalences of several notions of entropy, like 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 Gromov-hyperbolic metric spaces with convex geodesic bicombings.
We study locally compact, locally geodesically complete, locally CAT(k) spaces (GCBA(k)-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(k)-spaces satisfying a uniform packing condition at some fixed scale 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^k-complexes with bounded geometry.