Certain Homology Cycles of the Independence Complex of Grid Graphs

Let G be an infinite graph such that the automorphism group of G contains a subgroup K ∼= Z with the property that G/K is finite. We examine the homology of the independence complex Σ(G/I) of G/I for subgroups I of K of full rank, focusing on the case that G is the square, triangular, or hexagonal grid. Specifically, we look for a certain kind of homology cycles that we refer to as “cross-cycles”, the rationale for the terminology being that they are fundamental cycles of the boundary complex of some cross-polytope. For the special cases just mentioned, we determine the set Q(G) of rational numbers r such that there is a group I with the property that Σ(G/I) contains cross-cycles of dimension exactly r · |G/I | − 1; |G/I | denotes the size of the vertex set of G/I . In each of the three cases, Q(G) turns out to be an interval of the form [a, b] ∩ Q. For example, for the square grid, we obtain the interval [ 1 5 , 1 4 ] ∩ Q.