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Toral relative Cannon conjecture

Let (G,P)(G,P) be a relatively hyperbolic group pair, meaning that GG is relatively hyperbolic relative to the collection of subgroups PP. Assume that its Bowditch boundary is homeomorphic to S2S^2, that P≠∅P\neq\emptyset, and that every element of PP is Z2\mathbb{Z}^2. Toral relative Cannon conjecture. Then GG is Kleinian. This is a relative analogue of the Cannon conjecture motivated by the greater tractability of non-uniform lattices, whose peripheral subgroups are Z2\mathbb{Z}^2 and whose Bowditch boundary is a 22-sphere. The paper presents this as a relative version that may be more approachable, while the corresponding conjecture remains unresolved.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Toral Relative Cannon Conjecture

    Let (G,P)(G,P) be a relatively hyperbolic group pair with P≅Z2P \cong \mathbb{Z}^2, and suppose that its Bowditch boundary is homeomorphic to S2S^2. Toral Relative Cannon Conjecture. The group GG is virtually Kleinian. This is presented as a variation of Cannon's conjecture for relatively hyperbolic groups with toral peripheral subgroup; the source does not state whether it has been resolved.

    source: Darius Alizadeh, “Branched Covers of Hyperbolic Groups”, arXiv:2606.18086 (2026).

Sources & referencesView supporting material

Primary source

Daniel Groves, Peter Haïssinsky, Jason F. Manning, Damian Osajda, Alessandro Sisto and Genevieve S. Walsh, “Drilling hyperbolic groups”, arXiv:2406.14667 (2026).

3-manifoldsgeometric-group-theoryhyperbolic-groupsrelative-hyperbolicityrelatively-hyperbolic-groups

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