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

TL;DR

This paper develops a formula for the Farrell--Tate $K$-theory of groups with finite classifying spaces, applies it to $ ext{Out}(F_n)$, and explicitly computes the $p$-adic $K$-theory for certain cases, revealing new summands.

Contribution

It provides a general formula for $p$-adic Farrell--Tate $K$-theory using centralisers and applies it to $ ext{Out}(F_n)$, including explicit calculations for specific primes.

Findings

Computed the rational cohomology of centralisers in $ ext{Out}(F_{p+1})$.

Obtained explicit $p$-adic Farrell--Tate $K$-theory for $ ext{Out}(F_{p+1})$ with $p eq 3$.

Identified an infinite family of $ extbf{Q}_p$ summands in $K^1(B ext{Out}(F_n)) ensor_ ext{Z} extbf{Q}$.

Abstract

Using L\"uck'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 $p \geq 5$. 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 $K^1(B \text{Out}(F_{12})) \otimes_\mathbb{Z} \mathbb{Q}$.