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Tits Alternative in groups with proper product actions on proper Gromov-hyperbolic spaces

Jiaqi Cui, Renxing Wan

Abstract

In this paper, we study groups with property (PPH), i.e., there exist finitely many proper Gromov-hyperbolic spaces $X_1,\ldots, X_l$ on which $G$ acts cocompactly such that the diagonal action of $G$ on the $\ell^1$-product $\prod_{i=1}^lX_i$ is proper. We show that any finitely generated subgroup of a finitely generated group with property (PPH) either is amenable or contains $F_2$. Furthermore, we study groups with property (PPT), i.e., groups with property (PPH) so that $X_1,\cdots,X_l$ are all proper quasi-trees. We show that any finitely generated subgroup of a finitely generated group with property (PPT) either is virtually (locally-finite)-by-$\mathbb{Z}^n$ or contains $F_2$. Additionally, we establish that for a non-elementary hyperbolic group \(G\), \(G\) admits a proper diagonal action on a finite product of regular trees if and only if \(G\) has property (PPT). This result transforms a question posed by Button \cite{But19} into the problem of whether every non-elementary hyperbolic group has property (PPT).

Tits Alternative in groups with proper product actions on proper Gromov-hyperbolic spaces

Abstract

In this paper, we study groups with property (PPH), i.e., there exist finitely many proper Gromov-hyperbolic spaces on which acts cocompactly such that the diagonal action of on the -product is proper. We show that any finitely generated subgroup of a finitely generated group with property (PPH) either is amenable or contains . Furthermore, we study groups with property (PPT), i.e., groups with property (PPH) so that are all proper quasi-trees. We show that any finitely generated subgroup of a finitely generated group with property (PPT) either is virtually (locally-finite)-by- or contains . Additionally, we establish that for a non-elementary hyperbolic group , admits a proper diagonal action on a finite product of regular trees if and only if has property (PPT). This result transforms a question posed by Button \cite{But19} into the problem of whether every non-elementary hyperbolic group has property (PPT).

Paper Structure

This paper contains 15 sections, 31 theorems, 19 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Tits Alternative for groups with property (PPH)
  3. Tits Alternative for groups with property (PPT)
  4. From proper quasi-trees to regular trees
  5. Structure of the paper
  6. Preliminaries
  7. Gromov-hyperbolic spaces
  8. Amenable groups and Busemann quasimorphisms
  9. Tits Alternative
  10. Actions on quasi-lines
  11. Regular focal actions and confining subsets
  12. Tits Alternative in groups with property (PPH)
  13. Tits Alternative in groups with property (PPT)
  14. Property (QT)
  15. From proper quasi-trees to regular trees

Key Result

Theorem 1.1

Let $G$ be a finitely generated group with property (PPH). Then any finitely generated subgroup of $G$ is either amenable or contains a non-abelian free subgroup $F_2$. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: three different local characterizations of $T_1$ ($i\geq 0$).

Theorems & Definitions (61)

  • Theorem 1.1: Theorem \ref{['MainThm']}
  • Theorem 1.2: Proposition \ref{['Prop: NoPara']}
  • Remark 1.3
  • Corollary 1.4
  • Proposition 1.5: Proposition \ref{['Prop: AmeToEle']}
  • Theorem 1.6: Theorem \ref{['MainThm0']}
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Definition 2.4
  • ...and 51 more