Boundaries of Groups and Kleinian Groups — Problem 29

Definitions and mathematical statement

For a metric space $Y$, say $\operatorname{asdim}Y\leq n$ if for every $R>0$ there is a cover $\mathcal U$ of $Y$ whose members have uniformly bounded diameter and such that each ball of radius $R$ meets at most $n+1$ members. The diameter bound may depend on $R$, but not on the particular member of the cover. The asymptotic dimension is the smallest such $n$, or infinity if no finite $n$ works.

A CAT(0) group is a group $G$ acting properly and cocompactly by isometries on a proper geodesic CAT(0) space $X$, where triangles satisfy Euclidean upper comparison. Equip $G$ with a word metric from a finite generating set. Does every CAT(0) group satisfy \[ \operatorname{asdim}G<\infty? \] Asymptotic dimension is unchanged by quasi-isometry, so neither the finite generating set nor the choice of a geometric CAT(0) model changes the question.

The sought finite bound may depend on the group. No universal dimension bound for all CAT(0) groups is proposed, since free abelian groups already have arbitrarily large asymptotic dimension. The cover condition must hold at every large scale; a bounded-dimension triangulation or a finite-dimensional visual boundary alone is not the required conclusion. The problem concerns all proper CAT(0) models admitting geometric group actions, without assuming cubical, manifold or building structure.

Short English statement

Does every group acting geometrically on a CAT(0) space have finite asymptotic dimension?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 6200029 (AMR-061-0029), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005), Problem 29, PDF page 9. Original source indexed by AMR-061-0029. https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf.