Structure of geodesics in the Cayley graph of infinite Coxeter groups

Ryszard Szwarc. "Structure of geodesics in the Cayley graph of infinite Coxeter groups." Colloquium Mathematicae 95.1 (2003): 79-90. <http://eudml.org/doc/285326>.

@article{RyszardSzwarc2003,
abstract = {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.},
author = {Ryszard Szwarc},
journal = {Colloquium Mathematicae},
keywords = {Coxeter groups; geodesics; hyperbolic groups; Coxeter systems; presentations; Cayley graphs},
language = {eng},
number = {1},
pages = {79-90},
title = {Structure of geodesics in the Cayley graph of infinite Coxeter groups},
url = {http://eudml.org/doc/285326},
volume = {95},
year = {2003},
}

TY - JOUR
AU - Ryszard Szwarc
TI - Structure of geodesics in the Cayley graph of infinite Coxeter groups
JO - Colloquium Mathematicae
PY - 2003
VL - 95
IS - 1
SP - 79
EP - 90
AB - 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.
LA - eng
KW - Coxeter groups; geodesics; hyperbolic groups; Coxeter systems; presentations; Cayley graphs
UR - http://eudml.org/doc/285326
ER -