Packing and doubling in metric spaces with curvature bounded above

TL;DR

This paper investigates the geometric and measure-theoretic properties of GCBA(k)-spaces, establishing volume estimates, doubling conditions, and compactness results, with applications to M^k-complexes with bounded geometry.

Contribution

It provides new volume estimates, links doubling conditions to pure-dimensionality, and proves compactness results for GCBA(k)-spaces under packing and doubling assumptions.

Findings

Established a Croke-type local volume estimate depending only on dimension.

Proved that local doubling implies pure-dimensionality.

Demonstrated compactness of spaces under packing and doubling conditions.

Abstract

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.