Hyperbolic structures on groups

Abbott, Carolyn; Balasubramanya, Sahana; Osin, Denis

doi:10.2140/agt.2019.19.1747

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the Journal

Editorial Board

Editorial Interests

Subscriptions

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN (electronic): 1472-2739

ISSN (print): 1472-2747

Author Index

To Appear

Other MSP Journals

This article is available for purchase or by subscription. See below.

Hyperbolic structures on groups

Carolyn Abbott, Sahana H Balasubramanya and Denis Osin

Algebraic & Geometric Topology 19 (2019) 1747–1835

DOI: 10.2140/agt.2019.19.1747

Abstract

For every group $G$ , we define the set of hyperbolic structures on $G$ , denoted by $ℋ (G)$ , which consists of equivalence classes of (possibly infinite) generating sets of $G$ such that the corresponding Cayley graph is hyperbolic; two generating sets of $G$ are equivalent if the corresponding word metrics on $G$ are bi-Lipschitz equivalent. Alternatively, one can define hyperbolic structures in terms of cobounded $G$ –actions on hyperbolic spaces. We are especially interested in the subset $A ℋ (G) \subseteq ℋ (G)$ of acylindrically hyperbolic structures on $G$ , ie hyperbolic structures corresponding to acylindrical actions. Elements of $ℋ (G)$ can be ordered in a natural way according to the amount of information they provide about the group $G$ . The main goal of this paper is to initiate the study of the posets $ℋ (G)$ and $A ℋ (G)$ for various groups $G$ . We discuss basic properties of these posets such as cardinality and existence of extremal elements, obtain several results about hyperbolic structures induced from hyperbolically embedded subgroups of $G$ , and study to what extent a hyperbolic structure is determined by the set of loxodromic elements and their translation lengths.

PDF Access Denied

However, your active subscription may be available on Project Euclid at
https://projecteuclid.org/agt

We have not been able to recognize your IP address 3.235.78.122 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recommendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

Keywords

hyperbolic structures, group actions on hyperbolic spaces, acylindrical actions

Mathematical Subject Classification 2010

Primary: 20F65

Secondary: 20E08, 20F67

References

Publication

Received: 14 November 2017

Revised: 17 November 2018

Accepted: 22 December 2018

Published: 16 August 2019

Authors

Carolyn Abbott
University of California, Berkeley Berkeley, CA United States

Sahana H Balasubramanya
Department of Mathematics & Statistics University of North Carolina at Greensboro Greensboro, NC United States

Denis Osin
Department of Mathematics Vanderbilt University Nashville, TN United States

Volume 19, issue 4 (2019)