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Isometries of CAT(0) cube complexes are semi-simple

Isometries of CAT(0) cube complexes are semi-simple We consider an automorphism of an arbitrary CAT(0) cube complex. We study its combinatorial displacement and we show that either the automorphism has a fixed point or it preserves some combinatorial axis. It follows that when a f.g. group contains a distorted cyclic subgroup, it admits no proper action on a discrete space with walls. As an application Baumslag-Solitar groups and Heisenberg groups provide examples of groups having a proper action on measured spaces with walls, but no proper action on a discrete space with wall. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annales mathématiques du Québec Springer Journals

Isometries of CAT(0) cube complexes are semi-simple

Annales mathématiques du Québec , Volume 47 (2) – Oct 1, 2023

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References (23)

Publisher
Springer Journals
Copyright
Copyright © Fondation Carl-Herz and Springer Nature Switzerland AG 2021
ISSN
2195-4755
eISSN
2195-4763
DOI
10.1007/s40316-021-00186-2
Publisher site
See Article on Publisher Site

Abstract

We consider an automorphism of an arbitrary CAT(0) cube complex. We study its combinatorial displacement and we show that either the automorphism has a fixed point or it preserves some combinatorial axis. It follows that when a f.g. group contains a distorted cyclic subgroup, it admits no proper action on a discrete space with walls. As an application Baumslag-Solitar groups and Heisenberg groups provide examples of groups having a proper action on measured spaces with walls, but no proper action on a discrete space with wall.

Journal

Annales mathématiques du QuébecSpringer Journals

Published: Oct 1, 2023

Keywords: CAT(0) cube complexes; Spaces with walls; Classification of isometries; Baumslag–Solitar Groups; 20F65; 20F67; (51F15, 20F18)

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