The Reidemeister spectrum of 2-step nilpotent groups determined by graphs

TL;DR

This paper investigates the Reidemeister spectrum of 2-step nilpotent groups derived from graphs, introducing three methods to compute the spectrum based on graph structure, with applications to fixed point theory.

Contribution

It presents three novel methods for determining the Reidemeister spectrum of groups from graphs, extending analysis to small graphs and applications in topology.

Findings

Methods successfully applied to graphs with up to four vertices

Reidemeister spectra explicitly computed for several graph families

Applications demonstrated in topological fixed point theory

Abstract

In this paper we study the Reidemeister spectrum of 2-step nilpotent groups associated to graphs. We develop three methods, based on the structure of the graph, that can be used to determine the Reidemeister spectrum of the associated group in terms of the Reidemeister spectra of groups associated to smaller graphs. We illustrate our methods for several families of graphs, including all the groups associated to a graph with at most four vertices. We also apply our results in the context of topological fixed point theory for nilmanifolds.