Download this article
 Download this article For screen
For printing
Recent Issues

Volume 24
Issue 6, 2971–3570
Issue 5, 2389–2970
Issue 4, 1809–2387
Issue 3, 1225–1808
Issue 2, 595–1223
Issue 1, 1–594

Volume 23, 9 issues

Volume 22, 8 issues

Volume 21, 7 issues

Volume 20, 7 issues

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN 1472-2739 (online)
ISSN 1472-2747 (print)
Author Index
To Appear
 
Other MSP Journals
Strongly shortcut spaces

Nima Hoda

Algebraic & Geometric Topology 24 (2024) 3291–3325
Abstract

We define the strong shortcut property for rough geodesic metric spaces, generalizing the notion of strongly shortcut graphs. We show that the strong shortcut property is a rough similarity invariant. We give several new characterizations of the strong shortcut property, including an asymptotic cone characterization. We use this characterization to prove that asymptotically CAT (0) spaces are strongly shortcut. We prove that if a group acts metrically properly and coboundedly on a strongly shortcut rough geodesic metric space then it has a strongly shortcut Cayley graph and so is a strongly shortcut group. Thus we show that CAT (0) groups are strongly shortcut.

To prove these results, we use several intermediate results which we believe may be of independent interest, including what we call the circle tightening lemma and the fine Milnor–Schwarz lemma. The circle tightening lemma describes how one may obtain a quasi-isometric embedding of a circle by performing surgery on a rough Lipschitz map from a circle that sends antipodal pairs of points far enough apart. The fine Milnor–Schwarz lemma is a refinement of the Milnor–Schwarz lemma that gives finer control on the multiplicative constant of the quasi-isometry from a group to a space it acts on.

Keywords
strong shortcut property, asymptotically CAT(0) group, nonpositively curved group, geometric group theory
Mathematical Subject Classification
Primary: 20F65, 20F67, 51F30
References
Publication
Received: 16 October 2020
Revised: 14 November 2022
Accepted: 1 December 2022
Published: 7 October 2024
Authors
Nima Hoda
Department of Mathematics
Cornell University
Ithaca, NY
United States

Open Access made possible by participating institutions via Subscribe to Open.