Relative Cannon's Conjecture

Establish that every group hyperbolic relative to a finite collection of virtually \(\mathbb Z^2\) subgroups, with Bowditch boundary homeomorphic to the standard 2-sphere, is commensurable with the fundamental group of a finite-volume hyperbolic 3-manifold.

Background

The paper identifies the relative version of Cannon's Conjecture as a natural extension of the hyperbolic-group problem. The conjecture concerns a relatively hyperbolic group whose peripheral subgroups are virtually Z2\mathbb Z^2 and whose Bowditch boundary is a topological 2-sphere.

The stated conjecture predicts that such a group is commensurable with the fundamental group of a finite-volume hyperbolic 3-manifold. The paper proves a related conditional result: when the Bowditch boundary is Ahlfors regular and its Ahlfors regular conformal dimension is attained, the group is Kleinian and the peripheral subgroups are virtually Z2\mathbb Z^2; the full conjecture remains unresolved in the text.

References

An analogue of Cannon's Conjecture for relatively hyperbolic groups was proposed in Problem~57.

Let $G$ be a group hyperbolic relative to a collection ${H_1,\ldots,H_k}$ of virtually $\mathbb{Z}2$ subgroups. If the Bowditch boundary of $G$ is homeomorphic to $\mathbb{S}2$, then $G$ is commensurable with the fundamental group of a finite-volume hyperbolic $3$-manifold.

— On 2-Sphere Bowditch Boundaries Attaining Conformal Dimension  (2608.28346 - Pal et al., 28 Aug 2026) in Section 1, Introduction