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Algebraically hyperbolic groups

Giles Gardam, Dawid Kielak, Alan D. Logan

Abstract

We initiate the study of torsion-free algebraically hyperbolic groups; these groups generalise torsion-free hyperbolic groups and are intricately related to groups with no Baumslag--Solitar subgroups. Indeed, for groups of cohomological dimension $2$ we prove that algebraic hyperbolicity is equivalent to containing no Baumslag--Solitar subgroups. This links algebraically hyperbolic groups to two famous questions of Gromov; recent work has shown these questions to have negative answers in general, but they remain open for groups of cohomological dimension $2$. We also prove that algebraically hyperbolic groups are CSA, and so have canonical abelian JSJ-decompositions. In the two-generated case we give a precise description of the form of these decompositions.

Algebraically hyperbolic groups

Abstract

We initiate the study of torsion-free algebraically hyperbolic groups; these groups generalise torsion-free hyperbolic groups and are intricately related to groups with no Baumslag--Solitar subgroups. Indeed, for groups of cohomological dimension we prove that algebraic hyperbolicity is equivalent to containing no Baumslag--Solitar subgroups. This links algebraically hyperbolic groups to two famous questions of Gromov; recent work has shown these questions to have negative answers in general, but they remain open for groups of cohomological dimension . We also prove that algebraically hyperbolic groups are CSA, and so have canonical abelian JSJ-decompositions. In the two-generated case we give a precise description of the form of these decompositions.
Paper Structure (16 sections, 50 theorems, 25 equations, 1 figure)

This paper contains 16 sections, 50 theorems, 25 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Algebraically hyperbolic groups
  3. Main definition
  4. First examples
  5. Adian-free groups
  6. $\operatorname{BS}$-free groups of cohomological dimension $2$
  7. The CSA property
  8. Relationships between the three definitions
  9. AH versus weakly AH groups
  10. Poison subgroups
  11. Residually and fully residually (weakly) AH groups
  12. Two-generator algebraically hyperbolic groups
  13. Free products with amalgamation
  14. HNN-extensions
  15. Abelian splittings
  16. ...and 1 more sections

Key Result

Theorem 1

We have the following chain of proper inclusions of classes of groups:

Figures (1)

  • Figure 2.1: The graph $\Gamma_4$

Theorems & Definitions (102)

  • Definition 1.3
  • Theorem 1: Theorem \ref{['thm:inclusions']}
  • Theorem 2: Theorem \ref{['thm:coHom2Main']}
  • Theorem 3: Theorem \ref{['thm:LI']}
  • Theorem 4: Theorem \ref{['thm:CSA']}
  • Corollary 5: Corollary \ref{['corol:codimCSA']}
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Example 2.3
  • ...and 92 more