A remark on the connectedness of spheres in Cayley graphs

Antoine Gournay

doi:10.1016/j.crma.2014.05.005

Group theory/Topology

A remark on the connectedness of spheres in Cayley graphs
[Une remarque sur la connexité des sphères dans les graphes de Cayley]

Présenté par : the Editorial Board

Antoine Gournay ¹

¹ Université de Neuchâtel, rue Émile-Argand 11, CH-2000 Neuchâtel, Switzerland

Comptes Rendus. Mathématique, Volume 352 (2014) no. 7-8, pp. 573-576.

Résumé
Abstract

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 $r > 0$ tel que, pour tout $n \geq 0$ , l'intersection de la boule (dans un graphe de Cayley) de rayon $n + r$ 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 $B_{n + r}$ , the ball of radius $n + r$ , with $B_{n}^{c, \infty}$ , 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 $B_{n + r} \cap B_{n}^{c, \infty}$ is connected (for any n).

Reçu le : 2014-01-21
Accepté le : 2014-05-20
Publié le : 2014-06-05

DOI : 10.1016/j.crma.2014.05.005

Affiliations des auteurs :

Antoine Gournay ¹

¹ Université de Neuchâtel, rue Émile-Argand 11, CH-2000 Neuchâtel, Switzerland

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Antoine Gournay. A remark on the connectedness of spheres in Cayley graphs. Comptes Rendus. Mathématique, Volume 352 (2014) no. 7-8, pp. 573-576. doi : 10.1016/j.crma.2014.05.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.05.005/

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