1-Uryson width and covers

Abstract

We investigate the following question: Do there exist Riemannian polyhedra $X$ such that the 1-Uryson width of their universal covers ${\mathrm{UW}}_1(\tilde{X})$ is bounded but ${\mathrm{UW}}_1(X)$ is arbitrarily large? We rule out two specific cases: when $\ddot{I}{€}_1(X)$ 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 $\tilde{X}$. Precisely: $${\mathrm{UW}}_1(X)\le 6\cdot {\mathrm{UW}}_1(\tilde{X}).$$ We show that if $X$ is a Riemannian surface with boundary then $${\mathrm{UW}}_1(X)\le {\mathrm{UW}}_1(\tilde{X}).$$ Furthermore, we show that if there exist spaces $X$ for which ${\mathrm{UW}}_1(\tilde{X})$ 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.

27 pages. Corrects a gap in the proof of Theorem C. Includes other minor revisions following the referee's comments. To appear in Mathematische Zeitschrift