The Tits alternative for finite rank median spaces

Preprint English OPEN
Fioravanti, Elia;
  • Subject: Mathematics - Group Theory | Mathematics - Metric Geometry | Mathematics - Geometric Topology
    arxiv: Astrophysics::Cosmology and Extragalactic Astrophysics

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 ... View more
  • References (34)
    34 references, page 1 of 4

    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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