Questions in Geometric Group Theory — Q 2.11

Definitions and mathematical statement

A proper metric space has compact closed bounded balls. A geodesic metric space is CAT$(0)$ if its geodesic triangles satisfy Euclidean comparison: distances between side points are no greater than the corresponding comparison distances. Let a finitely generated discrete group $G$ act on a proper CAT$(0)$ space $X$ by isometries. Require proper discontinuity in the compact-set sense: \[ \{g\in G:gK\cap K\ne\varnothing\}\text{ is finite} \quad\text{for every compact }K\subset X. \] In particular, every point stabilizer is finite.

(a) Can $G$ be an infinite torsion group, meaning that every element has finite order although the group itself is infinite?

(b) If the action is also cocompact, meaning $X/G$ is compact, can $G$ contain an infinite torsion subgroup $H$? The subgroup in (b) is not assumed finitely generated, and its restricted action need not be cocompact.

Both questions permit arbitrarily large finite orders; torsion does not mean that a single exponent annihilates every element. The space may have infinite dimension, but it remains proper. A finite torsion group poses no obstruction and is not the requested example. Conversely, constructing an infinite torsion action with an infinite stabilizer, or on a nonproper space, would fail the stated hypotheses. The issue in the second part is the subgroup structure of a group acting cocompactly, rather than whether that ambient group itself has only torsion elements.

Short English statement

Can infinite torsion occur in a proper CAT(0) action, either as the whole finitely generated acting group or as a subgroup under cocompactness?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1100211 (AMR-010-0211), 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 2.11 (PDF page 8). Original source indexed by AMR-010-0211. https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf.