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arXiv · 1401.5249

A remark on the connectedness of spheres in Cayley graphs

Antoine Gournay
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Abstract

The aim of this small note is to prove an elementary yet useful properties of finitely presented groups. Let G be a finitely generated group with one end. Fix a (finite) generating set and let $B_n$ be the ball of radius $n$ around $e$. Let $B_n^{c,\infty}$ be the infinite connected component of the complement of $B_n$. Then G has connected spheres if there exists a $r >0$ such that $B_{n+r} \cap B_n^{c,\infty}$ is connected for all $n \geq 0$. This note shows that if G is finitely presented then it has connected spheres.

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BibTeXRIS

Antoine Gournay. 2014-08-12. A remark on the connectedness of spheres in Cayley graphs. https://doi.org/10.1016/j.crma.2014.05.005

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