Biggs tree groups

TL;DR

This paper extends Biggs' construction of finite permutation groups with large girth Cayley graphs, showing that with at least three colors, the groups contain the full alternating group, creating infinite families with specific properties.

Contribution

It demonstrates that for three or more colors, the constructed groups include the full alternating group, expanding the class of known groups with large girth Cayley graphs.

Findings

Groups contain the full alternating group for at least 3 colors

Constructs infinite families of groups with large girth Cayley graphs

Provides explicit permutation generators for these groups

Abstract

Biggs gave an explicit construction, using finite colored trees, of finite permutation groups whose Cayley graphs have valence \(C\) and girth tending to infinity as the radius \(R\) of the tree tends to infinity. We show that when the number of colors is at least 3, the group so presented contains the full alternating group on the vertices of the tree. This gives, for each \(C\geq 3\), an infinite family of pairs \((G_{C,R},S_{C,R})\) such that \(G_{C,R}\) is an alternating or symmetric group, \(S_{C,R}\) is a generating set of \(G_{C,R}\) of size \(C\) with an explicit permutation description of its generators, and such that the sequence of Cayley graphs \(\mathrm{Cay}(G_{C,R},S_{C,R})\) has constant valence \(C\) and girth tending to infinity as \(R\) tends to infinity.