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 $B_n$ 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 $n\ge 0$. This note shows that if G is finitely presented then it has connected spheres.

5p., 1 figure

References in corpus (4)

• Vanishing of the first reduced cohomology with values in an $L^p$-representation
• Strongly t-logarithmic t-generating sets: Geometric properties of some soluble groups
• Functions conditionally of negative type on groups acting on regular trees
• Absence of non-constant harmonic functions with ${\ell }^p$-gradient in some semi-direct products

Cited by in corpus (4)

• Mixing, malnormal subgroups and cohomology in degree one
• Limit theorems for random walks on Fuchsian buildings and Kac-Moody groups
• Absence of non-constant harmonic functions with ${\ell }^p$-gradient in some semi-direct products
• Connectedness of spheres in Cayley graphs