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Rank-one isometries of CAT(0) cube complexes and their centralisers

  • Anthony Genevois EMAIL logo
Published/Copyright: July 8, 2021
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Advances in Geometry
From the journal Advances in Geometry Volume 21 Issue 3

Abstract

If G is a group acting geometrically on a CAT(0) cube complex X and if g ∈ G has infinite order, we show that exactly one of the following situations occurs: (i) g defines a rank-one isometry of X; (ii) the stable centraliser SCG(g) = {h ∈ G ∣ ∃ n ≥ 1, [h, gn] = 1} of g is not virtually cyclic; (iii) FixY(gn) is finite for every n ≥ 1 and the sequence (FixY(gn)) takes infinitely many values, where Y is a cubical component of the Roller boundary of X which contains an endpoint of an axis of g. We also show that (iii) cannot occur in several cases, providing a purely algebraic characterisation of rank-one isometries.

Keywords: CAT(0) space; cube complex; rank-one isometry
MSC 2010: 20F65

  1. Communicated by: R. Weiss

  2. Funding: This work was supported by a public grant as part of the Fondation Mathématique Jacques Hadamard.

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Received: 2019-08-03
Revised: 2019-12-08
Revised: 2019-12-28
Published Online: 2021-07-08
Published in Print: 2021-07-27

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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  2. On vector bundles over reducible curves with a node
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https://doi.org/10.1515/advgeom-2021-0011
Keywords for this article
CAT(0) space; cube complex; rank-one isometry

Articles in the same Issue

  1. Frontmatter
  2. On vector bundles over reducible curves with a node
  3. Inscribed rectangle coincidences
  4. 𝔽p2-maximal curves with many automorphisms are Galois-covered by the Hermitian curve
  5. Tetrahedral cages for unit discs
  6. When is M0,n(ℙ1,1) a Mori dream space?
  7. Rank-one isometries of CAT(0) cube complexes and their centralisers
  8. Wall-crossing in genus-zero hybrid theory
  9. The curve Yn = Xℓ(Xm + 1) over finite fields II
  10. Uniqueness results for bodies of constant width in the hyperbolic plane
  11. A note on large Kakeya sets
  12. On static manifolds and related critical spaces with cyclic parallel Ricci tensor
  13. The motivic Igusa zeta function of a space monomial curve with a plane semigroup
  14. Real hypersurfaces of non-flat complex space forms with two generalized conditions on the Jacobi structure operator
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