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Automorphisms of relatively hyperbolic groups and the Farrell-Jones conjecture.

Naomi Andrew1,2, Yassine Guerch3,4, Sam Hughes1,5

  • 1Mathematical Institute, University of Oxford, Andrew Wiles Building, Observatory Quarter, OX2 6GG Oxford, UK.

Mathematische Annalen
|June 18, 2026
PubMed
Summary

The fibred Farrell-Jones conjecture (FJC) is proven for suspensions of hyperbolic and relatively hyperbolic groups. This work advances algebraic topology and group theory, with implications for automorphism groups and group extensions.

Keywords:
20E0820F6520F67Primary 18F25Secondary 20F28

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Area of Science:

  • Algebraic Topology
  • Geometric Group Theory
  • Homological Algebra

Background:

  • The Farrell-Jones conjecture (FJC) is a fundamental problem in algebraic topology concerning the algebraic K-theory of group rings.
  • Relatively hyperbolic groups and their suspensions form an important class of groups with rich geometric and algebraic properties.
  • JSJ decompositions provide a powerful tool for understanding the structure of relatively hyperbolic groups.

Purpose of the Study:

  • To prove the fibred Farrell-Jones conjecture (FJC) for a broad class of suspensions of relatively hyperbolic groups.
  • To extend the results to all suspensions of one-ended hyperbolic groups.
  • To explore applications of the proven FJC, including its validity for automorphism groups and its behavior under group extensions.

Main Methods:

  • Utilizing techniques from algebraic topology and geometric group theory.
  • Applying JSJ decomposition methods to analyze the structure of relatively hyperbolic groups.
  • Developing and employing methods specific to the fibred version of the Farrell-Jones conjecture.

Main Results:

  • The fibred Farrell-Jones conjecture is established for large classes of suspensions of relatively hyperbolic groups.
  • The conjecture is proven for all suspensions of one-ended hyperbolic groups.
  • Key applications include proving the FJC for automorphism groups of certain hyperbolic groups and demonstrating closure under specific extensions.

Conclusions:

  • The study significantly advances the understanding of the Farrell-Jones conjecture in the context of hyperbolic and relatively hyperbolic groups.
  • The results provide new insights into the structure of these groups via JSJ decompositions.
  • This work has broad implications for algebraic topology and group theory research.