Asymptotic dimension for covers with controlled growth

TL;DR

This paper introduces a relative version of asymptotic dimension, $\mathrm{asdim}_{\mathcal{M}}$ and its uniform variant $\overline{\mathrm{asdim}}_{\mathcal{M}}$, for classes of metric families $\mathcal{M}$ with growth controls, and proves their monotonicity under regular maps. Using these invariants, the authors derive strong non-embedding results between spaces of exponential growth and products with subexponential factors, including $\mathbb{H}^n \to \mathbb{H}^{n-1}\times Y$ ($n\ge 3$), $(T_3)^n \to (T_3)^{n-1}\times Y$ with bounded-degree $Y$, and $F^n \to \mathbb{Z}\wr F^{n-1}$. They establish lower bounds via exponentially distorted subsets and connect these to hyperbolic rank and subexponential corank, while providing upper bounds for subexponential-fibre situations through explicit decompositions of hyperbolic spaces and their products. A key outcome is a tight bound for $\overline{\mathrm{asdim}}_{\mathrm{poly}(d)}(\mathbb{H}^2)=2$ for all $d$, which yields new obstructions to regular maps from $\mathbb{H}^2$ into various product spaces, including those with subexponentially growing factors. Overall, the results enrich coarse-geometry tools for obstructing coarse embeddings and sharpen the landscape of large-scale dimensional invariants in spaces with controlled growth.

Abstract

We prove various obstructions to the existence of regular maps (or coarse embeddings) between commonly studied spaces. For instance, there is no regular map (or coarse embedding) $\mathbb H^n\to\mathbb H^{n-1}\times Y$ for $n\geq 3$, or $(T_3)^n \to (T_3)^{n-1}\times Y$ whenever $Y$ is a bounded degree graph with subexponential growth, where $T_3$ is the $3$-regular tree. We also resolve a question of Benjamini-Schramm-Timár, proving that there is no regular map $\mathbb H^2 \to T_3 \times Y$ whenever $Y$ is a bounded degree graph with at most polynomial growth, and no quasi-isometric embedding whenever $Y$ has subexponential growth. Finally, we show that there is no regular map $F^n\to \mathbb Z\wr F^{n-1}$ where $F$ is the free group on two generators. To prove these results, we introduce and study generalizations of asymptotic dimension which allow unbounded covers with controlled growth. For bounded degree graphs, these invariants are monotone with respect to regular maps (hence coarse embeddings).

Figures (1)

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Theorems & Definitions (102)

• Theorem A
• Theorem B
• Theorem C
• Corollary 1.2
• Corollary 1.3
• Proposition 1.4
• Proposition 1.5
• Theorem D
• Proposition 1.8
• Theorem E