The Chabauty space of closed subgroups of the three-dimensional Heisenberg group

Abstract

When equipped with the natural topology first defined by Chabauty, the closed subgroups of a locally compact group G form a compact space C{script}(G). We analyse the structure of C{script}(G) for some low-dimensional Lie groups, concentrating mostly on the 3-dimensional Heisenberg group H. We prove that C{script}(H) is a 6-dimensional space that is path-connected but not locally connected. The lattices in H form a dense open subset L{script}(H) ⊂ C{script}(H) that is the disjoint union of an infinite sequence of pairwise homeomorphic aspherical manifolds of dimension six, each a torus bundle over (S3 \T) × R, where T denotes a trefoil knot. The complement of L{script}(H) in C{script}(H) is also described explicitly. The subspace of C{script}(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{script}(H).