Boundaries of Groups and Kleinian Groups — Problem 13

Definitions and mathematical statement

An ideal boundary records asymptotic directions in a space with a specified geometry. For a proper geodesic hyperbolic space, its Gromov boundary can be represented by geodesic rays modulo finite Hausdorff distance, with the boundary topology. For a word-hyperbolic group one uses a Cayley graph. Other spaces, for example CAT(0) spaces, may instead use a visual boundary whose conventions must be fixed. The source does not restrict all its examples to one boundary construction.

Research problem (Tadeusz Januszkiewicz). Find further universality phenomena: classes of groups or spaces of different nature whose ideal boundaries are nevertheless all homeomorphic, beyond the examples involving Menger compacta, right-angled hyperbolic buildings, and Davis--Vinberg complexes.

A precise form of a proposed example would give a class $\mathcal C$, define $\partial_\infty X$ for its members, identify a compactum $B$, and prove \[ \partial_\infty X\cong B\qquad\text{for every }X\in\mathcal C. \] The source seeks genuinely further examples but supplies no criterion for "different nature," such as nonisomorphism or non-quasi-isometry, and no exhaustive target class. Neither criterion is added. Universality here means a common boundary homeomorphism type across a class; it is not automatically the separate assertion that every compactum embeds into that boundary.

The supplied summary mentions right-angled Coxeter examples with Menger or spherical boundaries and Menger-curve boundaries for certain random-group models, but gives no hypotheses or citations. Its phrase about "dimension-free compacta" does not specify a valid dimensional embedding assertion and is not imposed as part of this problem. These reports provide tentative examples/progress rather than a complete classification or a precise remaining conjecture.

Short English statement

Find new, structurally different families of groups or spaces sharing one ideal-boundary homeomorphism type, beyond the listed Menger/building/Davis--Vinberg examples. The boundary construction, diversity criterion, and scope are open-ended; common boundary type is distinguished from universal embedding properties.

Sources

• Local UnsolvedMath registry, permanent ID 6200013, source code catalogue entry, "Boundaries of Groups and Kleinian Groups --- Problem 13"; proposer: Tadeusz Januszkiewicz, proposed year 2005. Status partially solved, classification PARTIAL-PROGRESS, updated 2026-08-06.
• Kapovich, Problems on Boundaries of Groups and Kleinian Groups (2005), Problem 13, PDF page 5, https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf. Reported PDF-text-normalized extraction and access: 2026-07-29. The source labels it a problem; current status NEEDS_REVIEW. Rights: public author-hosted PDF; statement-reuse terms NEEDS_REVIEW.
• The statement names Menger compacta, right-angled hyperbolic buildings, and Davis--Vinberg complexes. The summary additionally mentions right-angled Coxeter and random-group examples without primary references or exact assumptions.