Let $M$ be a closed hyperbolic 3-manifold. One can use hyperbolic structure on $M$ to define hyperbolic volume $Vol(M)$. Thanks to Mostow's rigidity theorem the volume depends only on the topology of $M.$ Volume Conjecture gives a way to compute this invariant starting with any triangulation, by looking at the asymptotical behavior of Turaev-Viro invariants. These invariants are denoted $TV_n(M,q)$ and depend on the root of unity $q=e^{\frac{2\pi i}{n}}.$ They can be computed very explicitly for any triangulated $3-$manifold and are based on so-called $6j-$symbols for quantum group $U_q(sl_2)$. Here is the exact statement: $$ \lim_{n\longrightarrow \infty} \frac{2\pi i}{n}log\left ( TV_n(M,q) \right)=Vol(M). $$

Hyperbolic volume is an invariant of motivic origin. Namely, one can associate to every hyperbolic 3-manifold a motivic cohomology class in $H_{\mathcal{M}}^{1}(F, \mathbb{Q}(2))$ for some number field $F$. Hyperbolic volume $Vol(M)$ is obtained by applying Borel's regulator to it. It will be interesting to formulate an analog of the Volume Conjecture in some other motivic realization.

Question 1 One can look at other regulators, related to other cohomology theories, for instance $p-$adic regulator. So, probably, there exists a $p-$adic version of hyperbolic volume. Has it been studied?

Question 2 Is it possible to formulate an analog of the volume conjecture in this setting? For instance, substituting representations of $U_q(sl_2)$ with some similarly behaving category in characteristic $p?$

Question 3 If the answer to both questions is "yes", is it possible to come up with a formulation of the Volume conjecture in some context, where analytical difficulties of proving convergence are less severe?

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    $\begingroup$ Can you provide a reference for p-adic regulators? $\endgroup$
    – Ian Agol
    May 3 '19 at 3:07
  • $\begingroup$ I think, I had in mind something like the one defined here: arxiv.org/pdf/0707.3682.pdf . $\endgroup$ May 4 '19 at 16:19
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    $\begingroup$ An important point in the classical story is the fact that $\mathbb{CP}^1$ is the boundary of hyperbolic 3-space $\mathbb{H}^3$. This leads to relations between homology of the isometry group of $\mathbb{H}^3$ and K-theory and subsequently to the relation between the regulator and hyperbolic volumes. The $p$-adic analogue of that story of the hyperbolic space could be an interesting starting point for investigation. $\endgroup$ Jul 14 '19 at 18:34

I don't know how to answer your question, since I don't know about motives or $p$-adic regulators (a reference would be helpful). I'll just point out one possible relation which may just be a curiosity.

Catalan's constant ${\displaystyle G=\beta (2)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{2}}}={\frac {1}{1^{2}}}-{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}-{\frac {1}{7^{2}}}+{\frac {1}{9^{2}}}-\cdots }$ gives the volume of an ideal hyperbolic simplex associated to the right-isosceles triangle. Since the Whitehead link may be triangulated by four of these ideal tetrahedra, one knows that its volume is $4G = 3.66...$.

The series for $G$ also converges 2-adically, and Frank Calegari has shown that the 2-adic version of $G$ is irrational, something that does not appear to be known for $G$ itself. I have no idea if this is something like a 2-adic regulator for the hyperbolic manifold though.

  • $\begingroup$ Dear Ian, thanks for a beautiful remark! $\endgroup$ May 4 '19 at 16:16

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