1-Uryson width and covers

TL;DR

This paper explores the relationship between the 1-Uryson width of Riemannian polyhedra and their universal covers, establishing bounds for specific classes and suggesting limitations on possible counterexamples.

Contribution

It proves bounds on the 1-Uryson width for polyhedra with virtually cyclic fundamental groups and surfaces, and discusses the existence of spaces with bounded universal cover width but arbitrarily large base width.

Findings

For virtually cyclic fundamental groups, UW_1(X) ≤ 6 * UW_1(˜X).

For Riemannian surfaces with boundary, UW_1(X) ≤ UW_1(˜X).

Potential counterexamples must be low-dimensional, possibly among Riemannian 2-complexes.

Abstract

We investigate the following question: Do there exist Riemannian polyhedra $X$ such that the 1-Uryson width of their universal covers $\mathrm{UW}_1(\widetilde{X})$ is bounded but $\mathrm{UW}_1(X)$ is arbitrarily large? We rule out two specific cases: when $\pi_1(X)$ is virtually cyclic and when $X$ is a Riemannian surface. More specifically, we show that if $X$ is a compact polyhedron with a virtually cyclic fundamental group, then its 1-Uryson width is bounded by the 1-Uryson width of its universal cover $\widetilde{X}$. Precisely: $$\mathrm{UW}_1(X) \leq 6 \cdot \mathrm{UW}_1(\widetilde{X}).$$ We show that if $X$ is a Riemannian surface with boundary then $$\mathrm{UW}_1(X) \leq \mathrm{UW}_1(\widetilde{X}).$$ Furthermore, we show that if there exist spaces $X$ for which $\mathrm{UW}_1(\widetilde{X})$ is bounded while $\mathrm{UW}_1(X)$ is arbitrarily large, then such examples must already appear in low dimensions. In particular, such $X$ can be found among Riemannian $2$-complexes.