publication . Article . Other literature type . Preprint . 2019

The Tits alternative for finite rank median spaces

Fioravanti, Elia;
Open Access
  • Published: 23 May 2019 Journal: L’Enseignement Mathématique, volume 64, pages 89-126 (issn: 0013-8584, Copyright policy)
  • Publisher: European Mathematical Publishing House
Abstract
We prove a version of the Tits alternative for groups acting on complete, finite rank median spaces. This shows that group actions on finite rank median spaces are much more restricted than actions on general median spaces. Along the way, we extend to median spaces the Caprace-Sageev machinery and part of Hagen's theory of unidirectional boundary sets.
Subjects
arXiv: Astrophysics::Cosmology and Extragalactic Astrophysics
free text keywords: Combinatorics, Tits alternative, Mathematics, Mathematics - Group Theory, Mathematics - Geometric Topology, Mathematics - Metric Geometry
Related Organizations
34 references, page 1 of 3

2.2. Products. Given median spaces X1; X2, we can consider the product X1 X2, which is a median space itself with the `1 metric, i.e.

where the first two pieces are transverse and the third is null. Since h 2 H0 and k 2 H n (H0 [ H ) this partition is nontrivial. Finally, observe that H0 is inseparable and, thus, measurable. Corollary 2.9 now violates the irreducibility of X.

with ah ( h0 and bk ( k0; in particular, h; ah ; k; bk form a facing 4-tuple.

Set w := fh; h g 2 W and := h \ ah \ k \ bk X. Claim. For every nontrivial, reduced word u in a and b, if u = au0, we have u ah ; if u = a 1u0, we have u h; if u = bu0, we have u bk ; if u = b 1u0, we have u k.

[Bow15b] Brian H. Bowditch. Median and injective metric spaces. Preprint, 2015.

[Bow16] Brian H. Bowditch. Some properties of median metric spaces. Groups Geom. Dyn., 10(1):279-317, 2016. [OpenAIRE]

[BW12] Nicolas Bergeron and Daniel T. Wise. A boundary criterion for cubulation. Amer. J. Math., 134(3):843-859, 2012.

[CDH10] Indira Chatterji, Cornelia Druţu, and Frédéric Haglund. Kazhdan and Haagerup properties from the median viewpoint. Adv. Math., 225(2):882-921, 2010.

[CFI16] Indira Chatterji, Talia Fernós, and Alessandra Iozzi. The median class and superrigidity of actions on CAT(0) cube complexes. J. Topol., 9(2):349-400, 2016. With an appendix by Pierre-Emmanuel Caprace.

[CL10] Pierre-Emmanuel Caprace and Alexander Lytchak. At infinity of finitedimensional CAT(0) spaces. Math. Ann., 346(1):1-21, 2010.

[Cor13] Yves Cornulier. Group actions with commensurated subsets, wallings and cubings. arXiv:1302.5982v2, 2013.

[CS11] Pierre-Emmanuel Caprace and Michah Sageev. Rank rigidity for CAT(0) cube complexes. Geom. Funct. Anal., 21(4):851-891, 2011.

[Dil50] R. P. Dilworth. A decomposition theorem for partially ordered sets. Ann. of Math. (2), 51:161-166, 1950. [OpenAIRE]

[DP16] Thomas Delzant and Pierre Py. Cubulable Kähler groups. arXiv:1609.08474v1, 2016.

[Fio17a] Elia Fioravanti. Roller boundaries for median spaces and algebras. Preprint, 2017. [OpenAIRE]

34 references, page 1 of 3
Powered by OpenAIRE Open Research Graph
Any information missing or wrong?Report an Issue
publication . Article . Other literature type . Preprint . 2019

The Tits alternative for finite rank median spaces

Fioravanti, Elia;