Alexandrov spaces with large volume growth

TL;DR

This paper investigates n-dimensional Alexandrov spaces with nonnegative Ricci curvature-like conditions, demonstrating that sufficiently large volume growth implies these spaces have finite topological type, extending classical Riemannian geometry results.

Contribution

It generalizes classical Riemannian results to Alexandrov spaces with large volume growth, establishing finite topological type under certain conditions.

Findings

Large volume growth implies finite topological type

Extension of Riemannian geometric arguments to Alexandrov spaces

Conditions for nonnegative Ricci curvature in Alexandrov spaces

Abstract

Let $(X,d)$ be an $n$-dimensional Alexandrov space whose Hausdorff measure $\mathcal{H}^n$ satisfies a condition giving the metric measure space $(X,d,\mathcal{H}^n)$ a notion of having nonnegative Ricci curvature. We examine the influence of large volume growth on these spaces and generalize some classical arguments from Riemannian geometry showing that when the volume growth is sufficiently large, then $(X,d,\mathcal{H}^n)$ has finite topological type.