  We use assembly maps to study $\mathbf{TC}(\mathbb{A}[G];p)$, the topological
cyclic homology at a prime $p$ of the group algebra of a discrete group $G$
with coefficients in a connective ring spectrum $\mathbb{A}$. For any finite
group, we prove that the assembly map for the family of cyclic subgroups is an
isomorphism on homotopy groups. For infinite groups, we establish
pro-isomorphism, (split) injectivity, and rational injectivity results, as well
as counterexamples to injectivity and surjectivity. In particular, for
hyperbolic groups and for virtually finitely generated abelian groups, we show
that the assembly map for the family of virtually cyclic subgroups is injective
but in general not surjective.
