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Is the fundamental group of any compact hyperbolic 3-manifold embeddable into a p-adic group?

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 \...
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