The algebraic K-theory of $k[{\mathrm{SL}}_2({\mathbb{F}}_q)]$

We compute via trace methods the higher algebraic $K$-theory of the group ring $k[{\mathrm{SL}}_2({\mathbb{F}}_q)]$, as well as the related groups ${\mathrm{PSL}}_2({\mathbb{F}}_q)$, ${\mathrm{PGL}}_2({\mathbb{F}}_q)$, and ${\mathrm{GL}}_2({\mathbb{F}}_q)$, where $k$ is a perfect field of characteristic $p$ and $q=p^r$. At the core of the computation is the algebraic $K$-theory of the group ring of the Sylow $p$-subgroup, $k[C_p^r]$, which we determine via a theorem of Lück–Reich–Rognes–Varisco on cyclic assembly for topological cyclic homology. In the process, we reprove the cyclic assembly result in the language of Nikolaus–Scholze, analyse assembly for smaller families of subgroups, and develop further tools for computing topological cyclic homology of group rings.