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

Abstract

We compute via trace methods the higher algebraic $K$-theory of the group ring $k[\operatorname{SL}_2(\mathbb{F}_q)]$, as well as the related groups $\operatorname{PSL}_2(\mathbb{F}_q)$, $\operatorname{PGL}_2(\mathbb{F}_q)$, and $\operatorname{GL}_2(\mathbb{F}_q)$, 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.