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...

Soit $G$ un groupe et $\tau$ un $G$-arbre. Dans cet article, nous supposons que $G$ ne se scinde pas comme amalgame $G=A{*}_{C}B$, ou HNN extension $G=A{*}_C$ au-dessus d’un groupe $C$ qui stabilise un segment de longueur $k$ dans $\tau (k\ge 2)$; si de plus $\tau$ ne contient pas de sous-arbre $G$-invariant, nous montrons que le nombre de sommets de $\tau /G$ est majoré par 12$kT$, où $T$ mesure la complexité d’une présentation de $G$.

We prove that if a group acts properly and cocompactly on a systolic complex, in whose 1-skeleton there is no isometrically embedded copy of the 1-skeleton of an equilaterally triangulated Euclidean plane, then the group is word-hyperbolic. This was conjectured by D. T. Wise.