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

Using Luck’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 Out(Fn). 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 Out(Fp+1) which does not lift to an order p element in Aut(Fp+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 Out(Fp+1) for any prime p ≥ 5. Our arguments provide an infinite family of Qp summands in K1 (B Out(Fn)) ⊗Z Q, with no need for computer calculations. The smallest value of p to which this applies is p = 11. In this case we obtain a Q11 summand in K1 (B Out(F12)) ⊗Z Q.