[Paper Review] The K-theoretic Farrell-Jones conjecture for CAT(0)-groups

Christian Wegner|arXiv (Cornell University)|Dec 15, 2010

Homotopy and Cohomology in Algebraic Topology4 references4 citations

TL;DR

This paper proves the K-theoretic Farrell-Jones conjecture with coefficients for CAT(0)-groups by introducing strong homotopy actions and strong transfer reducibility, enabling a controlled transfer map from a point to large closed balls in the CAT(0)-space. The key result establishes that the K-theoretic assembly map is an isomorphism for all integers m, resolving a long-standing conjecture for this class of groups.

ABSTRACT

We prove the K-theoretic Farrell-Jones conjecture with (twisted) coefficients for CAT(0)-groups.

Motivation & Objective

• To establish the K-theoretic Farrell-Jones conjecture with coefficients for CAT(0)-groups, a class of groups acting cocompactly and properly on finite-dimensional CAT(0)-spaces.
• To overcome the challenge that closed balls in CAT(0)-spaces do not carry a strict G-action, only a homotopy G-action, by introducing strong homotopy actions that encode higher homotopies.
• To generalize prior results on hyperbolic and CAT(0)-groups by extending the bijectivity of the K-theoretic assembly map to all degrees m ∈ ℤ.
• To define and utilize the obstruction category and a transfer map from a point to a metric space with controlled geometry, enabling the vanishing of K-theory of the obstruction category.

Proposed method

• Introduce strong homotopy actions as a framework to encode group actions on spaces up to higher homotopies, generalizing homotopy actions.
• Define strong transfer reducibility over the family of virtually cyclic subgroups, ensuring the existence of a transfer map from a point to a metric space with controlled geometry.
• Use large closed balls in the CAT(0)-space as the target metric space for the transfer map, leveraging their contractibility and G-action up to homotopy.
• Construct a transfer map in controlled algebra using chain complexes and inclusion maps, ensuring that morphisms vanish at large distances.
• Apply controlled algebra techniques to show that the K-theory of the obstruction category vanishes, implying the assembly map is an isomorphism.
• Prove that CAT(0)-groups are strongly transfer reducible over the family of virtually cyclic subgroups, enabling the general result.

Experimental results

Research questions

• RQ1 Can the K-theoretic Farrell-Jones conjecture with coefficients be established for CAT(0)-groups, given their lack of strict group actions on closed balls?
• RQ2 How can higher homotopies in group actions be systematically controlled to enable higher K-theory computations?
• RQ3 What conditions on a metric space allow the construction of a transfer map that induces an isomorphism on K-theory?
• RQ4 Is strong transfer reducibility a sufficient condition for the K-theoretic Farrell-Jones conjecture to hold?
• RQ5 Does the vanishing of the K-theory of the obstruction category imply the bijectivity of the K-theoretic assembly map?

Key findings

• The K-theoretic Farrell-Jones conjecture with coefficients holds for all CAT(0)-groups, extending previous results that only established bijectivity in degrees m ≤ 0 and surjectivity in degree m = 1.
• CAT(0)-groups are strongly transfer reducible over the family of virtually cyclic subgroups, a key technical condition enabling the proof.
• The obstruction category’s K-theory vanishes under the strong transfer reducibility condition, which implies the assembly map is an isomorphism.
• The transfer map from the point to large closed balls in the CAT(0)-space is constructed using controlled algebra and induces the identity on K-theory after composition with the projection.
• The proof establishes that the K-theoretic assembly map is an isomorphism for all m ∈ ℤ, resolving the conjecture in full generality for CAT(0)-groups.
• The result implies that products and free products of CAT(0)-groups also satisfy the K-theoretic Farrell-Jones conjecture with coefficients.