Digraphical Regular Representations of Infinite Finitely Generated Groups

AbstractA directed Cayley graphXis called a digraphical regular representation (DRR) of a groupGif the automorphism group ofXacts regularly onX. LetSbe a finite generating set of the infinite cyclic groupZ. We show that a directed Cayley graphX(Z,S) is aDRRofZif and only ifS≠S−1. IfX(Z,S) is not aDRRwe show thatAut(X(Z,S)) =D∞. As a general result we prove that a Cayley graphXof a finitely generated torsion-free nilpotent groupNis aDRRif and only if no non-trivial automorphism ofNof finite order leaves the generating set invariant

Which finitely generated Abelian groups admit isomorphic Cayley graphs?

We show that Cayley graphs of finitely generated Abelian groups are rather rigid. As a consequence we obtain that two finitely generated Abelian groups admit isomorphic Cayley graphs if and only if they have the same rank and their torsion parts have the same cardinality. The proof uses only elementary arguments and is formulated in a geometric language.Comment: 16 pages; v2: added reference, reformulated quasi-convexity, v3: small corrections; to appear in Geometriae Dedicat

Graphs, permutations and topological groups

Various connections between the theory of permutation groups and the theory of topological groups are described. These connections are applied in permutation group theory and in the structure theory of topological groups. The first draft of these notes was written for lectures at the conference Totally disconnected groups, graphs and geometry in Blaubeuren, Germany, 2007.Comment: 39 pages (The statement of Krophollers conjecture (item 4.30) has been corrected

The CI problem for infinite groups

Sherpa Romeo green journal: open accessA finite group G is a DCI-group if, whenever S and S0 are subsets of G with the Cayley graphs Cay(G,S) and Cay(G,S0) isomorphic, there exists an automorphism ϕ of G with ϕ(S) = S0. It is a CI-group if this condition holds under the restricted assumption that S = S−1. We extend these definitions to infinite groups, and make two closely-related definitions: an infinite group is a strongly (D)CIf-group if the same condition holds under the restricted assumption that S is finite; and an infinite group is a (D)CIf-group if the same condition holds whenever S is both finite and generates G. We prove that an infinite (D)CI-group must be a torsion group that is not locallyfinite. We find infinite families of groups that are (D)CIf-groups but not strongly (D)CIf-groups, and that are strongly (D)CIf-groups but not (D)CI-groups. We discuss which of these properties are inherited by subgroups. Finally, we completely characterise the locally-finite DCI-graphs on Zn. We suggest several open problems related to these ideas, including the question of whether or not any infinite (D)CIgroup exists.Ye