An application of the theory of FI -algebras to graph configuration spaces

Recent work of An et al. (Subdivisional spaces and graph braid groups. arXiv:1708.02351, 2019), as well as Ramos (Algebraic Geom Topol. arXiv:1609.05611, 2018), has shown that the homology groups of configuration spaces of graphs can be equipped with the structure of a finitely generated graded module over a polynomial ring. In this work we study this module structure in certain families of graphs using the language of FI-algebras recently explored by Nagel and Römer (FI- and OI-modules with varying coefficients. arXiv:1710.09247, 2017). As an application we prove that the syzygies of the modules in these families exhibit a range of stable behaviors.