On the K-theory of certain extensions of free groups

Abstract

Since $\operatorname*{Hol}(F_{n})$ embeds into $\operatorname*{Aut}(F_{n+1})$, one can construct inductively the subgroups ${\mathcal{H}}_{(n)}$ of $\operatorname*{Aut}(F_{n+1})$ by setting ${{\mathcal{H}}_{(1)}=\operatorname*{Hol}(F_{2})}$ and ${{\mathcal{H}}_{(n)}=F_{n+1}\rtimes{\mathcal{H}}_{(n-1)}}$. We show that the FJCw holds for ${\mathcal{H}}_{(n)}$. Moreover, we calculate the lower K-theory for the groups ${\mathcal{H}}_{(n)}$.