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?

• Can you provide a reference for p-adic regulators? May 3 '19 at 3:07
• I think, I had in mind something like the one defined here: arxiv.org/pdf/0707.3682.pdf . May 4 '19 at 16:19
• 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. 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.