Equivariant bifurcations in $4$-dimensional fixed point spaces

TL;DR

This paper explores new infinite series of finite groups acting on 4-dimensional spaces that serve as counterexamples to the Ize conjecture, and investigates their associated bifurcations within the context of Lie group structures.

Contribution

It introduces novel infinite series of finite groups as counterexamples to the Ize conjecture and examines their bifurcation behavior using Lie group frameworks.

Findings

New infinite series of finite groups countering the Ize conjecture.

Identification of Lie groups containing these finite groups.

Insights into bifurcation behaviors related to these groups.

Abstract

In this paper we continue the study of group representations which are counterexamples to the Ize conjecture. As in the previous papers by Lauterbach [14] and Lauterbach & Matthews [15] we find new infinite series of finite groups leading to such counterexamples. These new series are quite different from the previous ones, for example the group orders do not form an arithmetic progression. However, as before we find Lie groups which contain all these groups. This additional structure was observed, but not used in the previous studies of this problem. Here we also investigate the related bifurcations. To a large extent, these are closely related to the presence of mentioned compact Lie group containing the finite groups. This might give a tool to study the bifurcations related to all low dimensional counterexamples of the Ize conjecture. It also gives an indication of where we can expect to find examples where the bifurcation behavior is different from what we have seen in the known examples.