Asymptotic-Type Dimension Bounds through Combinatorial Approaches

TL;DR

This paper introduces a probabilistic framework to establish sharp asymptotic dimension bounds in metric geometry, resolving longstanding questions for manifolds with nonnegative Ricci curvature and extending results to metric-measure spaces.

Contribution

It provides new combinatorial and probabilistic methods to derive sharp asymptotic dimension bounds, including for Riemannian manifolds and metric measure spaces, with applications to geometric analysis.

Findings

Proves that for metric measure spaces with volume doubling constant C_D, asdim_{AN}(X) ≤ ⌊log_2 C_D⌋.

Shows that complete Riemannian manifolds with nonnegative Ricci curvature have asdim ≤ their dimension n.

Extends polynomial growth bounds for asymptotic dimension from graphs to metric-measure spaces.

Abstract

We develop a probabilistic framework for large-scale dimension bounds in metric geometry, based on padded decompositions, randomized ball carving on net graphs, and the Lov\'asz Local Lemma. For metric measure spaces with volume doubling constant $C_{\mathsf D}$, we prove the sharp bound $\mathrm{asdim}_{AN}(X)\le \mathrm{dim}_{AN}(X)\le \lfloor{\log_2 C_{\mathsf D}}\rfloor$. In particular, if $(M,g)$ is a complete Riemannian $n$-manifold with $\mathrm{Ric}_g\ge 0$, then $\mathrm{asdim}(M)\le n$, thereby settling a question of Papasoglu on manifolds with nonnegative Ricci curvature. We also show that if $(X,\mathsf{d},\mathfrak{m})$ is proper, volume noncollapsed, and has polynomial volume growth rate $\rho^V(X)$, then $\mathrm{asdim}(X)\le \lfloor{\rho^V(X)}\rfloor$. Moreover, the corresponding control function can be chosen to have polynomial growth. This extends Papasoglu's sharp asymptotic-dimension bound from graphs of polynomial growth to a metric-measure setting. As applications, we study equality in the polynomial-growth bound for universal covers of nilmanifolds, and under nonnegative Ricci curvature we relate the equality case in the volume-doubling bound to Gromov largeness, obtaining in particular a consequence for complete manifolds with positive scalar curvature.