Boundaries of Groups and Kleinian Groups — Problem 6

Definitions and mathematical statement

A finitely generated group $G$ is hyperbolic relative to a finite family $\mathcal P=\{P_1,\ldots,P_s\}$ in the Bowditch sense if it acts on a connected fine hyperbolic graph with finitely many edge orbits and finite edge stabilizers, whose infinite vertex stabilizers are exactly conjugates of the $P_i$. Fine means that each edge belongs to only finitely many embedded circuits of any specified length; hyperbolic means that geodesic triangles are uniformly thin.

The Bowditch boundary is the compact boundary obtained from the graph's Gromov boundary by adjoining its infinite-valence parabolic vertices, with the usual observer topology. Its equivariant homeomorphism type is independent of the chosen graph for this relatively hyperbolic structure.

Does there exist such a group with nonempty peripheral family, each $P_i$ finitely generated nilpotent of class at least three, and \[ \partial_B(G,\mathcal P)\cong S^n \] for some $n\geq1$? Nilpotency class is defined using the lower central series $\gamma_1(P)=P$, $\gamma_{j+1}(P)=[P,\gamma_j(P)]$: class $c$ means $\gamma_{c+1}(P)=1$ but $\gamma_c(P)\ne1$.

The sphere is a topological sphere; no round metric or conformal action is required. The nilpotent groups must occur as peripheral subgroups of the given structure, rather than merely as arbitrary subgroups of $G$. Abelian and two-step nilpotent cusp groups do not meet the requested class threshold.

Short English statement

Can a spherical relatively hyperbolic boundary coexist with nilpotent peripheral groups of class at least three?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 6200006 (AMR-061-0006), 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 6, PDF page 3. Original source indexed by AMR-061-0006. https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf.