Boundaries of Groups and Kleinian Groups — Problem 25

Definitions and mathematical statement

Use the finite-dimensional $Z$-structure convention of the source. A pair $(\overline X,Z)$ is a $Z$-structure for a group $G$ if $\overline X$ is a compact Euclidean retract, $Z\subset\overline X$ is closed and homotopically negligible, and $X=\overline X\setminus Z$ admits a free proper cocompact $G$-action. Homotopically negligible means there is a homotopy $H_t:\overline X\to\overline X$ with $H_0$ the identity and $H_t(\overline X)\subset X$ for every $t>0$.

The nullity condition is also required: for each compact $K\subset X$ and each open cover of $\overline X$, all but finitely many translates $gK$ lie in a single member of that cover. An $EZ$-structure further requires that the action extend continuously to $\overline X$, hence to its boundary $Z$.

If $G$ is word-hyperbolic and $(\overline X,Z)$ is any such $EZ$-structure, must there be a homeomorphism \[ h:Z\longrightarrow\partial G\quad\text{with}\quad h(gz)=g\,h(z)? \] Here $\partial G$ is the Gromov boundary of a Cayley graph, defined by geodesic rays modulo bounded distance. The comparison is equivariant, not merely a comparison of cohomology or shape. The specified $EZ$ axioms, including nullity and extension of the action, are essential; an arbitrary compactification of a contractible cocompact $G$-space is not covered by the question.

Short English statement

Must every EZ-boundary of a hyperbolic group be equivariantly homeomorphic to its Gromov boundary?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 6200025 (AMR-061-0025), 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 25, PDF page 8. Original source indexed by AMR-061-0025. https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf.