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.

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