We prove that the continuous group cohomology groups of a locally profinite group G with coefficients in a smooth k-representation π of G are isomorphic to the Ext-groups Extⁱ_G(1,π) computed in the category of smooth k-representations of G. We apply this to show that if π is a supersingular representation of GL₂(Q_p) over an algebraic closure of F_p, then the continuous group cohomology of SL₂(Q_p) with values in π vanishes.
        
        Furthermore, we prove that the continuous group cohomology groups of a p-adic reductive group G, with coefficients in an admissible smooth representation of G over k are finitely generated over k and similarly, the continuous cohomology groups of G with coefficients in an admissible unitary Q_p-Banach space representation Π, are finite dimensional. We show that the continuous group cohomology of SL₂(Q_p) with values in non-ordinary irreducible Q_p-Banach space representations of GL₂(Q_p) vanishes.
        
        We then show that the continuous cohomology groups of a p-adic reductive group with coefficients in the locally analytic vectors of an admissible Q_p-Banach space representation are homeomorphic to those with coefficients in the Banach space representation itself. Moreover, we deduce that the canonical topologies on those continuous cohomology groups are Hausdorff and are the uniquely determined finest locally convex topologies.