On the Farrell--Tate $K$-theory of $\text{Out}(F_n)$

Using Lück’s Chern character isomorphism we obtain a general formula in terms of centralisers for the $p$-adic Farrell–Tate $K$-theory of any discrete group $G$ with a finite classifying space for proper actions. We apply this formula to $\text{Out}(F_n)$. The case $n=p+1$ turns out to be especially interesting for the following reason: Up to conjugacy there is exactly one order $p$ element in $\text{Out}(F_{p+1})$ which does not lift to an order $p$ element in $\text{Aut}(F_{p+1})$. We compute the rational cohomology of the centraliser of this element and as a consequence obtain a full calculation of the $p$-adic Farrell–Tate $K$-theory of $\text{Out}(F_{p+1})$ for any prime $p\ge 5$. Our arguments provide an infinite family of ${\mathbb{Q}}_p$ summands in $K^1(B\text{Out}(F_n)){\otimes }_{\mathbb{Z}}\mathbb{Q}$, with no need for computer calculations: the first such summand is in $K^1(B\text{Out}(F_{12})){\otimes }_{\mathbb{Z}}\mathbb{Q}$.