The Chabauty Space of Closed Subgroups of the Three-dimensional Heisenberg Group

When equipped with the natural topology first defined by Chabauty, the closed subgroups of a locally compact group G form a compact space C(G). We analyse the structure of C(G) for some low-dimensional Lie groups, concentrating mostly on the 3-dimensional Heisenberg group H. We prove that C(H) is a 6-dimensional space that is path–connected but not locally connected. The lattices in H form a dense open subset L(H) ⊂ C(H) that is the disjoint union of an infinite sequence of pairwise–homeomorphic aspherical manifolds of dimension six, each a torus bundle over (S r T ) × R, where T denotes a trefoil knot. The complement of L(H) in C(H) is also described explicitly. The subspace of C(H) consisting of subgroups that contain the centre Z(H) is homeomorphic to the 4–sphere, and we prove that this is a weak retract of C(H).