[Paper Review] Farrell-Jones Conjecture for free-by-cyclic groups

Mladen Bestvina, Koji Fujiwara+1 more|arXiv (Cornell University)|May 31, 2019

Geometric and Algebraic Topology11 references4 citations

TL;DR

This paper proves the Farrell-Jones Conjecture for free-by-cyclic groups using geometric group theory techniques, specifically train track maps for free group automorphisms and acylindrical group actions on trees. The key result establishes that all free-by-cyclic groups satisfy both the K- and L-theoretic Farrell-Jones Conjectures, extending known results for hyperbolic and CAT(0) groups.

ABSTRACT

We prove the Farrell-Jones conjecture for free-by-cyclic groups. The proof uses recently developed geometric methods for establishing the Farrell-Jones Conjecture.

Motivation & Objective

• To establish the Farrell-Jones Conjecture for free-by-cyclic groups, a class of groups that generalize free and hyperbolic groups.
• To extend the known range of groups satisfying the Farrell-Jones Conjecture by proving it for a new class with rich geometric and algebraic structure.
• To apply geometric methods—particularly acylindrical group actions and transverse covers of trees—to prove the conjecture in a non-trivial setting.
• To demonstrate that the class of groups satisfying the Farrell-Jones Conjecture is closed under certain extensions, using a transitivity principle.

Proposed method

• Utilizes train track maps for automorphisms of free groups to analyze the dynamics of the action on the associated Bass-Serre tree.
• Constructs a transverse covering of a simplicial tree by subtrees stabilized by subgroups isomorphic to free groups times Z, leveraging the structure of the mapping torus.
• Applies the notion of N-F-amenable actions to show that the group action on a compact Euclidean retract satisfies the geometric conditions required for the Farrell-Jones Conjecture.
• Employs the class AC(VNil) of groups built from nilpotent and free groups via successive F-amenable actions, showing that stabilizers of vertices in the skeleton of the transverse cover lie in this class.
• Uses the acylindricity of the group action on the skeleton of the transverse cover to control finite stabilizers and ensure the action is sufficiently regular.
• Applies the transitivity principle from Bartels-Lück to deduce the conjecture for extensions from the conjecture on the quotient and the base group.

Experimental results

Research questions

• RQ1 Does the Farrell-Jones Conjecture hold for free-by-cyclic groups, which are extensions of free groups by Z?
• RQ2 Can geometric group theoretic methods, such as train track maps and acylindrical actions on trees, be used to prove the conjecture for non-hyperbolic groups?
• RQ3 Is the class AC(VNil) closed under taking products and subgroups, and does it contain the stabilizers of the group action on the transverse cover?
• RQ4 Can the Farrell-Jones Conjecture be extended from a base group to a group extension when the base satisfies the conjecture and the kernel is free?
• RQ5 What conditions ensure that a group action on a space is finitely F-amenable, and how does this relate to the conjecture?

Key findings

• The Farrell-Jones Conjecture holds for all free-by-cyclic groups, both in K-theory and L-theory.
• The stabilizer of each edge in the transverse cover is isomorphic to a free group times Z, and hence lies in the class AC(VNil), which is closed under products and contains all groups satisfying the conjecture.
• The action of the free-by-cyclic group on the skeleton of the transverse cover is acylindrical, which ensures the necessary regularity for applying the geometric axioms.
• The group action on the compact Euclidean retract is finitely F-amenable, satisfying the key geometric condition for the conjecture to hold.
• The proof relies on the transitivity principle: if the quotient group satisfies the conjecture and the stabilizers of the action lie in AC(VNil), then the extension satisfies the conjecture.
• The result generalizes previous theorems for hyperbolic, CAT(0), and virtually solvable groups, showing that free-by-cyclic groups form a new class satisfying the conjecture.