Abstract

A group $G$ splits over a subgroup $C$ if $G$ is either a free product with amalgamation $A \underset{C}{\ast} B$ or an HNN-extension $G=A \underset{C}{\ast} (t)$. We invoke Bass-Serre theory and classify all infinite groups which admit cubic Cayley graphs of connectivity two in terms of splittings over a subgroup.

Highlights

  • The study of the structure of groups in terms of the connectivity of their Cayley graphs was started by Jung and Watkins

  • They characterized infinite transitive graphs of connectivity one whose automorphism groups act on their vertex sets as primitive and regular permutation groups [13]

  • A topic that has been already paired with the connectivity of Cayley graphs in order to study them is the planarity of infinite Cayley graphs

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Summary

Introduction

They characterized infinite transitive graphs of connectivity one whose automorphism groups act on their vertex sets as primitive and regular permutation groups [13]. Servatius characterized planar groups with low connectivity in terms of the fundamental group of the graph of groups Theorem 1.1 is a direct consequence of Theorems 4.3, 4.5, 5.4 and 5.8, where we discuss in detail the planarity of the corresponding Cayley graphs in each case, as well as their presentations This allows us to obtain as a corollary the results of [10], as well as full presentations for the non-planar groups with cubic Cayley graphs of connectivity two. By applying Bass–Serre theory, we naturally determine the structure of the group in terms of splitting avoiding the nuisance above, which was the way we originally obtained it

Preliminaries
General structure of the tree decomposition
Tree decomposition of Type I
Two Generators
Three Generators
Tree decomposition of Type II
Two generators
Three generators
Open Questions
Full Text
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