[Une remarque sur la connexité des sphères dans les graphes de Cayley]
L'objectif de cette note est de montrer qu'un groupe qui a un seul bout et est finiment présenté possède la propriété des sphères connexes. Cette propriété consiste à dire qu'il existe un tel que, pour tout , l'intersection de la boule (dans un graphe de Cayley) de rayon et de la composante infinie dans le complémentaire de la boule de rayon n est connexe.
The aim of this short note is to prove a useful result about the connectedness of spheres in Cayley graphs. By sphere, one refers to the sphere connected at infinity: the intersection of , the ball of radius , with , the infinite component ball of the complement of the ball of radius n. We show that in a finitely presented group with one end, there exists r such that is connected (for any n).
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@article{CRMATH_2014__352_7-8_573_0,
author = {Gournay, Antoine},
title = {A remark on the connectedness of spheres in {Cayley} graphs},
journal = {Comptes Rendus. Math\'ematique},
pages = {573--576},
publisher = {Elsevier},
volume = {352},
number = {7-8},
year = {2014},
doi = {10.1016/j.crma.2014.05.005},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2014.05.005/}
}
TY - JOUR AU - Gournay, Antoine TI - A remark on the connectedness of spheres in Cayley graphs JO - Comptes Rendus. Mathématique PY - 2014 SP - 573 EP - 576 VL - 352 IS - 7-8 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2014.05.005/ DO - 10.1016/j.crma.2014.05.005 LA - en ID - CRMATH_2014__352_7-8_573_0 ER -
%0 Journal Article %A Gournay, Antoine %T A remark on the connectedness of spheres in Cayley graphs %J Comptes Rendus. Mathématique %D 2014 %P 573-576 %V 352 %N 7-8 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2014.05.005/ %R 10.1016/j.crma.2014.05.005 %G en %F CRMATH_2014__352_7-8_573_0
Gournay, Antoine. A remark on the connectedness of spheres in Cayley graphs. Comptes Rendus. Mathématique, Tome 352 (2014) no. 7-8, pp. 573-576. doi : 10.1016/j.crma.2014.05.005. http://www.numdam.org/articles/10.1016/j.crma.2014.05.005/
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