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,∞), the infinite component ball of the complement of the ball of radiusn. We show that in a finitely presented group with one end, there exists rsuch that B_(n+r)∩B_n~(c,∞) is connected (for anyn).
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