Boundaries of Groups and Kleinian Groups — Problem 26

Definitions and mathematical statement

A $Z$-structure for a group $G$ is a compactification $\overline X=X\cup Z$ in which $\overline X$ is a compact Euclidean retract, $Z$ is a $Z$-set, and $G$ acts freely, properly and cocompactly on $X$. Being a $Z$-set means that the identity of $\overline X$ can be homotoped instantaneously into $X$. One also requires nullity: translates of each fixed compact subset of $X$ eventually become arbitrarily small with respect to every open cover of $\overline X$.

Use the explicit cell-like comparison in the source: two boundaries have a common compact metrizable domain mapping continuously and surjectively onto each, with both maps cell-like. A surjection is cell-like when every fiber has the shape of a point, meaning it contracts inside each neighborhood in an ambient absolute neighborhood retract.

Can one group $G$ have two $Z$-structure boundaries $Z_1,Z_2$ which are not cell-like equivalent? Equivalently, must every pair of boundaries admitted by these axioms have such a common domain and such maps?

The spaces giving the two structures may be different, and the boundary actions need not extend: extension is an additional $EZ$ condition, not part of this question. A homeomorphism would give cell-like equivalence, but the converse need not hold. Therefore examples of nonhomeomorphic boundaries do not settle the problem. The question concerns the stated stronger relation beyond the shape-equivalence information already available for $Z$-boundaries.

Short English statement

Can the same group have two Z-boundaries that remain different even up to cell-like equivalence?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 6200026 (AMR-061-0026), 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 26, PDF page 8. Original source indexed by AMR-061-0026. https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf.