The topology of local commensurability graphs

We initiate the study of the p-local commensurability graph of a group, where p is a prime. This graph has vertices consisting of all finite-index subgroups of a group, where an edge is drawn between A and B if [A : A boolean AND B] and [B : A boolean AND B] are both powers of p. We show that any component of the p-local commensurability graph of a group with all nilpotent finite quotients is complete. Further, this topological criterion characterizes such groups. In contrast to this result, we show that for any prime p the p-local commensurability graph of any large group (e.g. a nonabelian free group or a surface group of genus two or more or, more generally, any virtually special group) has geodesics of arbitrarily long length.