Homological algebra for nonarchimedean completions of group algebras

Let $V$ be a complete discrete valuation ring, and let $G$ be either a word-hyperbolic group or a reductive $p$-adic group. We prove that the canonical morphism $V[G]\to V[G{]}^{†}$ from the group algebra to its dagger completion is an isocohomological morphism (also known as an idempotent or a homotopy epimorphism). This allows us to deduce homological properties about the algebra $V[G{]}^{†}$ in terms of the relatively simpler group algebra of $G$. These properties are then used towards Hochschild and cyclic homology computations.