Questions in Geometric Group Theory — Q 1.9

Definitions and mathematical statement

Let $X_G$ be a connected finite two-dimensional CW complex with fundamental group $G$, equipped with a piecewise Euclidean length metric after a finite subdivision. Let $H\leq G$ be finitely presented, and let $X_H\to X_G$ be the connected covering corresponding to $H$, with the lifted metric. Write $\widetilde X$ for the universal cover. For $x\in X_H$, choose a lift $\widetilde x$ and define \[ I(x)=\frac12\inf\{d(\widetilde x,h\widetilde x):h\in H\setminus\{1\}\}, \] with infimum $+\infty$ if $H$ is trivial. This is the covering injectivity radius, equivalently half the infimal length of an essential loop based at $x$.

Assume $I(x)$ tends to infinity as $x$ leaves every compact subset of $X_H$: for each $R$ there is a compact $K_R$ outside which $I(x)>R$. Must the subgroup inclusion be a quasi-isometric embedding? Explicitly, for word metrics from finite generating sets, must constants $A\geq1$, $B\geq0$ exist such that \[ A^{-1}|h|_H-B\leq |h|_G\leq A|h|_H+B\qquad(h\in H)? \] The upper inequality follows from finite generation; the lower inequality is the substantive conclusion. The original question concerns finite presentations generally, with the word-hyperbolic ambient case singled out as particularly significant. Divergence is required uniformly outside compact sets, not just along a chosen ray.

Short English statement

Does the disappearance of short essential loops far out in a subgroup cover force the subgroup to be undistorted?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1100109 (AMR-010-0109), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Bestvina - Questions in Geometric Group Theory (pdf) (2004), Question 1.9 (PDF page 3). Original source indexed by AMR-010-0109. https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf.