Geodesic
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

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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.
DOI : 10.4064/cm95-1-7
Keywords: 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

Ryszard Szwarc 1

1 Institute of Mathematics Wrocław University Pl. Grunwaldzki 2/4 50-384 Wrocław, Poland 
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Ryszard Szwarc. Structure of geodesics in the Cayley graph of
 infinite Coxeter groups. Colloquium Mathematicum, Tome 95 (2003) no. 1, pp. 79-90. doi: 10.4064/cm95-1-7

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