TITLE: Is the fundamental group of any compact hyperbolic 3-manifold embeddable into a p-adic group?
QUESTION [12 upvotes]: Is it true that for every compact hyperbolic $3$-manifold $M$ there exists a prime $p$, a  finite field extension $K/\mathbb{Q}_p$, and an injective group homomorphism $$\tau \colon \pi_1(M) \to \mathrm{PSL}_2(\mathcal{O}_K) ?$$ 
Here $\mathcal{O}_K$ is the ring of integers of $K$.
This question is obviously related to the existence of faithful low dimensional representation of hyperbolic groups (see the following survey by Aschenbrenner, Friedl, and Wilton).
An affirmative solution will give a (partial) confirmation of Conjecture 1 from this paper by Luo.
A paper by Calegari and Emerton also discusses such group homomorphisms $\tau$.
Note that the answer is obviously 'yes' if $M$ is arithmetic.

REPLY [11 votes]: I think this follows easily from a couple of facts. The discrete faithful representation of an orientable hyperbolic 3-manifold leads to an embedding $\pi_1(M) \to PSL_2(S) \subset PSL_2(k)$, where $k$ is a number field, and $S$ is a finitely-generated subring (generated e.g. by the coefficients of matrices of generators). For most prime ideals $\mathcal{P} \subset \mathcal{O}_k$, $S$ will have non-negative valuation, and hence will lie in the valuation ring $R_{\mathcal{P}} \subset k_{\mathcal{P}}$ (the ring of $\mathcal{P}$-adic integers), hence $\pi_1(M) \hookrightarrow PSL_2(R_\mathcal{P})$. Such a field is a finite extension of $\mathbb{Q}_p$, where $\mathcal{P}|p$. (I'm attempting to follow the notation and results of Chapter 0.7 of MacLachlan-Reid). 
It's possible I've misinterpreted your notation. I'm interpreting your $K=k_\mathcal{P}, \mathcal{O}_K=R_\mathcal{P}$ (the valuation ring) to translate to MacLachlan-Reid's notation. If $\mathcal{O}_K$ does not denote the valuation ring, then I think you should clarify what you mean. This seems to me to be the only reasonable interpretation with $K=\mathbb{Q}_p, \mathcal{O}_K=\mathbb{Z}_p$.