A remark on the connectedness of spheres in Cayley graphs
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 Bn be the ball of radius n around e. Let Bnc,∞ be the infinite connected component of the complement of Bn. Then G has connected spheres if there exists a r >0 such that Bn+r Bnc,∞ is connected for all n ≥ 0. This note shows that if G is finitely presented then it has connected spheres.
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