Let (W,S) be a Coxeter system such that no two generators in S commute. Assume that the Cayley graph of (W,S) does not contain adjacent hexagons. Then for any two vertices x and y in the Cayley graph of W and any number k ≤ d = dist(x,y) there are at most two vertices z such that dist(x,z) = k and dist(z,y) = d - k. Allowing adjacent hexagons, but assuming that no three hexagons can be adjacent to each other, we show that the number of such intermediate vertices at a given distance from x and y is at most 3. This means that the group W is hyperbolic in a sense stronger than that of Gromov.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm95-1-7,
author = {Ryszard Szwarc},
title = {Structure of geodesics in the Cayley graph of infinite Coxeter groups},
journal = {Colloquium Mathematicae},
volume = {96},
year = {2003},
pages = {79-90},
zbl = {1061.20038},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm95-1-7}
}
Ryszard Szwarc. Structure of geodesics in the Cayley graph of infinite Coxeter groups. Colloquium Mathematicae, Tome 96 (2003) pp. 79-90. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm95-1-7/