The volume conjecture for simple generalized hyperbolic polyhedra

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Let PP) be a simple generalized hyperbolic polyhedron with dihedral angles θ1,…,θm\theta_1,\dots,\theta_m at the edges e1,…,eme_1,\dots,e_m, and 11-skeleton Γ\Gamma. Let colrcol_r be a sequence of rr-admissible colorings of the edges e1,…,eme_1,\dots,e_m of Γ\Gamma such that

2πlim⁡r→+∞colr(ek)r=π±θk.2\pi\lim_{r\rightarrow+\infty}\frac{col_r(e_k)}{r}=\pi\pm\theta_k.

The volume conjecture for polyhedra. At q=e2π−1rq=e^{\frac{2\pi\sqrt{-1}}{r}}, as rr runs over all the odd integers,

lim⁡r→+∞2πrln⁡∣⟨Γ,colr⟩q∣=Vol⁡(P).\lim_{r\rightarrow+\infty}\frac{2\pi}{r}\ln\big|\langle\Gamma,col_r\rangle_q\big|=\operatorname{Vol}(P).

This extends volume-conjecture formulations for simple hyperideal and compact polyhedra to simple generalized hyperbolic polyhedra, with regular, ideal, or hyperideal vertices. The surrounding discussion attributes related forms to earlier work, but the supplied text gives no resolution status.

References

Primary source

Giulio Belletti and Tian Yang, “Asymptotics of quantum 6j-symbols and generalized hyperbolic tetrahedra”, arXiv:2308.13864 (2023).

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