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ABSTRACT: In this talk, I will discuss the question as to determine geometric properties of orientable closed hyperbolic 3-manifolds by the finite quotients of their fundamental groups. I will explain an approach to match up the (Zariski-dense algebraic) PSL_2 representations of a pair of those groups, up to conjugacy, if there is an isomorphism between their profinite completions. Assuming the p-adic Borel regulator injectivity conjecture, one may deduce that the hyperbolic volume, the invariant quaternion algebra, and the arithmeticity are profinite properties.