Yue's critical exponent gap conjecture for complex hyperbolic Kleinian groups

From papers

Let mm be a positive integer, let HCm\mathbf{H}_{\mathbb{C}}^{m} be complex hyperbolic mm-space, and let Γ\Gamma be a convex cocompact isometry group of HCm\mathbf{H}_{\mathbb{C}}^{m}. The critical exponent of Γ\Gamma is denoted by δΓ\delta_{\Gamma}. Yue's conjecture. Suppose that Γ\Gamma is a convex cocompact isometry group of HCm\mathbf{H}_{\mathbb{C}}^{m}. Then either Γ\Gamma is a uniform lattice, in which case δΓ=2m\delta_{\Gamma}=2m, or

δΓ2m1.\delta_{\Gamma}\leq 2m-1.

The conjecture proposes a gap below the maximal critical exponent for convex cocompact groups that are not uniform lattices. The result discussed in the source produces counterexamples to an analogous gap in dimensions m=2m=2 and 33 for finitely generated discrete groups, but the supplied text does not establish whether Yue's stated convex cocompact conjecture itself is resolved.

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Primary source

Subhadip Dey and Beibei Liu, “Complex hyperbolic Kleinian groups of large critical exponents”, arXiv:2205.05127 (2023).

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