Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Naomi Andrew, Irakli Patchkoria

TL;DR

The paper develops a general p-adic Farrell–Tate K-theory framework for discrete groups with a finite model for proper actions, leveraging Lück's equivariant Chern character to reduce computations to centralisers. Applying this to Out(F_n) yields full Tate K-theory calculations in several n ranges around primes p, with a standout case n=p+1 where a unique order-$p$ element Φ (non-liftable to Aut(F_{p+1})) drives new odd K-theory phenomena. The authors establish an explicit rational lower bound for the p-adic Tate K-theory in Out(F_{p+1}) for p≥11, construct an infinite family of nontrivial K^1-summands in BOut(F_n) without computations, and provide detailed analyses of centralisers and their cohomology that govern the presence of odd Tate classes. Collectively, these results illuminate the interaction between group actions on Outer space, centraliser cohomology, and duality phenomena in p-adic K-theory, and connect to extensive prior work on Out(F_n) and related groups.

Abstract

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

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

TL;DR

The paper develops a general p-adic Farrell–Tate K-theory framework for discrete groups with a finite model for proper actions, leveraging Lück's equivariant Chern character to reduce computations to centralisers. Applying this to Out(F_n) yields full Tate K-theory calculations in several n ranges around primes p, with a standout case n=p+1 where a unique order- element Φ (non-liftable to Aut(F_{p+1})) drives new odd K-theory phenomena. The authors establish an explicit rational lower bound for the p-adic Tate K-theory in Out(F_{p+1}) for p≥11, construct an infinite family of nontrivial K^1-summands in BOut(F_n) without computations, and provide detailed analyses of centralisers and their cohomology that govern the presence of odd Tate classes. Collectively, these results illuminate the interaction between group actions on Outer space, centraliser cohomology, and duality phenomena in p-adic K-theory, and connect to extensive prior work on Out(F_n) and related groups.

Abstract

Using Lück's Chern character isomorphism we obtain a general formula in terms of centralisers for the -adic Farrell--Tate -theory of any discrete group with a finite classifying space for proper actions. We apply this formula to . The case turns out to be especially interesting for the following reason: Up to conjugacy there is exactly one order element in which does not lift to an order element in . We compute the rational cohomology of the centraliser of this element and as a consequence obtain a full calculation of the -adic Farrell--Tate -theory of for any prime . Our arguments provide an infinite family of summands in , with no need for computer calculations: the first such summand is in .

Paper Structure

This paper contains 23 sections, 17 theorems, 115 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries from homotopy theory
  3. Klein's dualising complex and the norm map
  4. Farrell--Tate K-theory and duality in complex K-theory
  5. Examples
  6. The general linear group SL3(Z)
  7. The general linear group GLp-1(Z) for a prime p at least 5
  8. The symplectic group Sp p-1(Z) for a prime p at least 5
  9. The mapping class group for a prime p at least 5
  10. Amalgamated products of finite groups
  11. Elements of order p and their centralisers
  12. The p-adic Farrell--Tate K-theory of Out(Fn) for n=p-1,p,p+1,p+2,p+3.
  13. The case n=p-1
  14. The case n=p
  15. The case n=p+1
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $G$ be a discrete group with a finite $\underline{E}G$. Then for any $m\in \mathbb{Z}$, the equivariant Chern character map is an isomorphism, where $\mathop{\mathrm{con}}\nolimits_p(G)$ denotes the set of conjugacy classes of the non-trivial $p$ power order elements, and $C \langle g \rangle$ denotes the centraliser of the cyclic subgroup generated by $g$.

Figures (4)

  • Figure 1:
  • Figure 2: Graphs with isometries realising $R_l$ and $\theta_{st}$
  • Figure 3: The outer automorphisms considered in Proposition \ref{['prop:nice_representative']} are all realised by a graph automorphism rotating this $\Gamma$ by $2\pi/p$.
  • Figure 4: Schematic for the Whitehead move described in Lemma \ref{['lem:equivariant_slide']}.

Theorems & Definitions (37)

  • Theorem 1.1: Lück
  • Proposition A
  • Theorem B
  • Proposition 3.1
  • Definition 3.2: Klein2001
  • Definition 3.3
  • Example 3.4
  • Proposition 3.5
  • proof
  • Remark 3.6
  • ...and 27 more