## Structure of geodesics in the Cayley graph of infinite Coxeter groups

### Volume 95 / 2003

Colloquium Mathematicum 95 (2003), 79-90
MSC: Primary 20F55, 20F05.
DOI: 10.4064/cm95-1-7

#### 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 $k\le d={\rm dist}(x,y)$ there are at most two vertices $z$ such that ${\rm dist}(x,z)=k$ and ${\rm dist}(z,y)=d-k$. 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.