In this paper we will show that finitely generated torsion-free 2-step nilpotent groups of Hirsch length at most 6 do not have the R∞-property, while there are examples of such groups of Hirsch length 7 that do have the R∞-property. * Supported by Methusalem grant METH/21/03 -long term structural funding of the Flemish Government. † Researcher funded by FWO PhD-fellowship fundamental research (file number: 1102424N).

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.