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](https://arxiv.org/pdf/1205.0202.pdf)).


An affirmative solution will give a (partial) confirmation of Conjecture 1 from this [paper by Luo](https://arxiv.org/pdf/1201.2228.pdf).


A [paper by Calegari and Emerton](http://www.math.uchicago.edu/~fcale/papers/Fritz.pdf) also discusses such group homomorphisms $\tau$.


Note that the answer is obviously 'yes' if $M$ is arithmetic.



Is the fundamental group of any compact hyperbolic 3-manifold embeddable into a p-adic group?
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](https://link.springer.com/book/10.1007/978-1-4757-6720-9)). 


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 $K=\mathbb{Q}\_p, \mathcal{O}\_K=\mathbb{Z}\_p$. 


