Structure of geodesics in the Cayley graph of infinite Coxeter groups

Ryszard Szwarc¹

¹ Institute of Mathematics Wrocław University Pl. Grunwaldzki 2/4 50-384 Wrocław, Poland

Colloquium Mathematicum, Tome 95 (2003) no. 1, pp. 79-90.

Résumé

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 $k\le d=\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(x,y)$ there are at most two vertices $z$ such that $\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(x,z)=k$ and $\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(z,y)=d-k$. 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.

Détail

DOI : 10.4064/cm95-1-7

Mots-clés : coxeter system generators commute assume cayley graph does contain adjacent hexagons vertices cayley graph number dist there vertices dist dist d k allowing adjacent hexagons assuming three hexagons adjacent each other number intermediate vertices given distance means group hyperbolic sense stronger gromov