Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Uniform undistortion from barycentres, and applications to hierarchically hyperbolic groups

Carolyn Abbott, Mark Hagen, Harry Petyt, Abdul Zalloum

TL;DR

The paper establishes that groups acting uniformly properly on injective metric spaces have uniformly undistorted infinite cyclic subgroups, and extends this to hierarchically hyperbolic groups (HHGs) by analyzing projections to hyperbolic spaces in the HHG structure. It develops barycentre techniques in injective spaces to produce uniform quasi-axes and uses the HHG distance formula to prove a uniform lower bound on translation lengths after a uniform power, giving a quantified form of translation discreteness for HHGs. The authors demonstrate sharpness via central extensions and quasimorphisms, constructing HHG structures that are not $\mathfrak S$-translation discrete and explaining the limitations of uniform bounds in non-proper settings. These results impact the understanding of translation-length spectra, uniform exponential growth in HHGs, and the structure of abelian subgroups, while raising several open questions about translation discreteness across different HHG structures and central extensions.

Abstract

We show that infinite cyclic subgroups of groups acting uniformly properly on injective metric spaces are uniformly undistorted. In the special case of hierarchically hyperbolic groups, we use this to study translation lengths for actions on the associated hyperbolic spaces. We then use quasimorphisms to produce examples where these latter results are sharp.

Uniform undistortion from barycentres, and applications to hierarchically hyperbolic groups

TL;DR

The paper establishes that groups acting uniformly properly on injective metric spaces have uniformly undistorted infinite cyclic subgroups, and extends this to hierarchically hyperbolic groups (HHGs) by analyzing projections to hyperbolic spaces in the HHG structure. It develops barycentre techniques in injective spaces to produce uniform quasi-axes and uses the HHG distance formula to prove a uniform lower bound on translation lengths after a uniform power, giving a quantified form of translation discreteness for HHGs. The authors demonstrate sharpness via central extensions and quasimorphisms, constructing HHG structures that are not -translation discrete and explaining the limitations of uniform bounds in non-proper settings. These results impact the understanding of translation-length spectra, uniform exponential growth in HHGs, and the structure of abelian subgroups, while raising several open questions about translation discreteness across different HHG structures and central extensions.

Abstract

We show that infinite cyclic subgroups of groups acting uniformly properly on injective metric spaces are uniformly undistorted. In the special case of hierarchically hyperbolic groups, we use this to study translation lengths for actions on the associated hyperbolic spaces. We then use quasimorphisms to produce examples where these latter results are sharp.
Paper Structure (21 sections, 36 theorems, 36 equations, 3 figures)

This paper contains 21 sections, 36 theorems, 36 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Translation discreteness in Cayley graphs
  3. Non-proper actions arising from hierarchical hyperbolicity
  4. Outline of the paper
  5. Barycentres and translation lengths
  6. Background on hierarchical hyperbolicity
  7. Proof of Theorem \ref{['thm:main']}
  8. Step 1: Quasi-axes
  9. Step 2: Choosing which point to move
  10. Step 3: Organising distance formula terms
  11. Step 4: Controlling orthogonal terms
  12. Step 5: Accumulating distance in nested domains
  13. S--translation discreteness
  14. Positive examples
  15. Quasimorphisms, central extensions, and bounded classes
  16. ...and 6 more sections

Key Result

Theorem 1.2

Let $G$ be a group acting uniformly properly on an injective metric space $X$. Infinite-order elements of $G$ are uniformly undistorted. Moreover, every infinite-order $g\in G$ admits an $\varepsilon$--quasi-axis in $X$ for every $\varepsilon>0$.

Figures (3)

  • Figure 1: The properties of the domain $V\sqsubsetneq U$ constructed in Lemma \ref{['lem:V']}.
  • Figure 2: Finding the domain $V\sqsubsetneq U$.
  • Figure 3: A schematic of the sets $g^i\rho^V_U=\rho^{g^iV}_U$ when $i\in \mathcal{I}$ in $\mathcal{C}U$.

Theorems & Definitions (80)

  • Theorem 1.2
  • Corollary 1.3: HHGs are translation discrete
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 70 more