Boundaries of Groups and Kleinian Groups — Problem 93

Definitions and mathematical statement

A hyperbolic group $\Gamma$ is a finitely generated group with a Gromov-hyperbolic Cayley graph; its ideal boundary is $\partial_\infty\Gamma$. In a usual representation-theoretic setting, a homomorphism $\rho:\Gamma\to H$ into a semisimple Lie group is studied relative to a parabolic subgroup $P\leq H$ and the flag space $H/P$. A candidate boundary map has the equivariance property \[ \xi(\gamma z)=\rho(\gamma)\xi(z),\qquad \xi:\partial_\infty\Gamma\to H/P. \] An Anosov structure requires additional contraction/transversality data; a continuous equivariant boundary map alone is not its definition.

The source asks to extend "this relation" between Anosov structure and dynamics on a limit set to representations of other hyperbolic groups. The antecedent relation, the original group class, the target group, the parabolic/flag space, the limit-set definition, and the dynamical hypotheses are not supplied. Consequently the task is to generalize a missing source theorem or characterization, rather than to prove an identifiable equivalence from the extract alone. The flag-space setting above is explanatory context and is not substituted for that missing relation.

Short English statement

Generalize the source's relation between Anosov structures and limit-set dynamics to other hyperbolic groups. The extracted record omits the relation and the target representation data needed to state a precise theorem.

Sources

• Misha Kapovich, Problems on Boundaries of Groups and Kleinian Groups (2005), Problem 93, PDF page 23, attributed to Anna Wienhard, https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf; accessed 2026-07-29, PDF-text-normalized extraction.
• The supplied research summary mentions Labourie and Guichard--Wienhard on Anosov representations, without individual publication identifiers or a precise statement of the relation to be extended.
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $6200093$, source code catalogue entry, Boundaries of Groups and Kleinian Groups --- Problem 93. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.