The University of Southampton
University of Southampton Institutional Repository

Quasi-isometric diversity of marked groups

Quasi-isometric diversity of marked groups
Quasi-isometric diversity of marked groups
We use basic tools of descriptive set theory to prove that a closed set S of marked groups has 2ℵ0 quasi-isometry classes provided every non-empty open subset of S contains at least two non-quasi-isometric groups. It follows that every perfect set of marked groups having a dense subset of finitely presented groups contains 2ℵ0 quasiisometry classes. These results account for most known constructions of continuous families of non-quasi-isometric finitely generated groups. We use them to prove the existence of 2ℵ0 quasi-isometry classes of finitely generated groups having interesting algebraic, geometric, or model-theoretic properties (e.g., such groups can be torsion, simple, verbally complete or they can all have the same elementary theory).
1753-8416
488-503
Minasyan, Ashot
3de640f5-d07b-461f-b130-5b1270bfdb3d
Osin, D.
712badf4-f382-4fec-851b-92f076d38c11
Witzel, S.
c0c39111-5312-44cc-b5c9-e7bdedfbc1a8
Minasyan, Ashot
3de640f5-d07b-461f-b130-5b1270bfdb3d
Osin, D.
712badf4-f382-4fec-851b-92f076d38c11
Witzel, S.
c0c39111-5312-44cc-b5c9-e7bdedfbc1a8

Minasyan, Ashot, Osin, D. and Witzel, S. (2021) Quasi-isometric diversity of marked groups. Journal of Topology, 14 (2), 488-503.

Record type: Article

Abstract

We use basic tools of descriptive set theory to prove that a closed set S of marked groups has 2ℵ0 quasi-isometry classes provided every non-empty open subset of S contains at least two non-quasi-isometric groups. It follows that every perfect set of marked groups having a dense subset of finitely presented groups contains 2ℵ0 quasiisometry classes. These results account for most known constructions of continuous families of non-quasi-isometric finitely generated groups. We use them to prove the existence of 2ℵ0 quasi-isometry classes of finitely generated groups having interesting algebraic, geometric, or model-theoretic properties (e.g., such groups can be torsion, simple, verbally complete or they can all have the same elementary theory).

Text
qi-25 - Accepted Manuscript
Download (417kB)
Text
- Accepted Manuscript
Restricted to Repository staff only
Request a copy

More information

Accepted/In Press date: 25 February 2021
Published date: June 2021

Identifiers

Local EPrints ID: 447318
URI: http://eprints.soton.ac.uk/id/eprint/447318
ISSN: 1753-8416
PURE UUID: d6020ad5-efee-41be-8701-30c2ae3ae052
ORCID for Ashot Minasyan: ORCID iD orcid.org/0000-0002-4986-2352

Catalogue record

Date deposited: 09 Mar 2021 17:31
Last modified: 17 Mar 2024 03:12

Export record

Contributors

Author: Ashot Minasyan ORCID iD
Author: D. Osin
Author: S. Witzel

Download statistics

Downloads from ePrints over the past year. Other digital versions may also be available to download e.g. from the publisher's website.

View more statistics

Atom RSS 1.0 RSS 2.0

Contact ePrints Soton: eprints@soton.ac.uk

ePrints Soton supports OAI 2.0 with a base URL of http://eprints.soton.ac.uk/cgi/oai2

This repository has been built using EPrints software, developed at the University of Southampton, but available to everyone to use.

We use cookies to ensure that we give you the best experience on our website. If you continue without changing your settings, we will assume that you are happy to receive cookies on the University of Southampton website.

×