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Automorphisms of relatively hyperbolic groups and the Farrell--Jones Conjecture

Naomi Andrew, Yassine Guerch, Sam Hughes

Abstract

We prove the fibred Farrell--Jones Conjecture (FJC) in $A$-, $K$-, and $L$-theory for a large class of suspensions of relatively hyperbolic groups, as well as for all suspensions of one-ended hyperbolic groups. We deduce two applications: (1) FJC for the automorphism group of a one-ended group hyperbolic relative to virtually polycyclic subgroups; (2) FJC is closed under extensions of FJC groups with kernel in a large class of relatively hyperbolic groups. Along the way we prove a number of results about JSJ decompositions of relatively hyperbolic groups which may be of independent interest.

Automorphisms of relatively hyperbolic groups and the Farrell--Jones Conjecture

Abstract

We prove the fibred Farrell--Jones Conjecture (FJC) in -, -, and -theory for a large class of suspensions of relatively hyperbolic groups, as well as for all suspensions of one-ended hyperbolic groups. We deduce two applications: (1) FJC for the automorphism group of a one-ended group hyperbolic relative to virtually polycyclic subgroups; (2) FJC is closed under extensions of FJC groups with kernel in a large class of relatively hyperbolic groups. Along the way we prove a number of results about JSJ decompositions of relatively hyperbolic groups which may be of independent interest.
Paper Structure (23 sections, 51 theorems, 15 equations)

This paper contains 23 sections, 51 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Applications
  3. Remarks on the proofs
  4. Fixed and Periodic Subgroups, the classes AC(VNIL) versus FJC, and localising invariants
  5. Structure of the paper
  6. Background on the Farrell--Jones conjecture
  7. Free Products of Groups and their Automorphisms
  8. Free products of groups
  9. Growth under an automorphism of a free product
  10. Actions on trees and JSJ decompositions
  11. Tree of cylinders
  12. JSJ decompositions of one-ended relatively hyperbolic groups
  13. The periodic JSJ decomposition
  14. Trees associated with an automorphism of a one-ended relatively hyperbolic group
  15. The Periodic JSJ tree
  16. ...and 8 more sections

Key Result

Theorem 2.1

The class $\mathop{\mathrm{\mathbf{FJC}_\mathbf{X}}}\nolimits$ is closed under the following operations:

Theorems & Definitions (101)

  • Remark
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4: Bartels
  • proof
  • Theorem 2.5: Knopf
  • ...and 91 more