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(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 (S³ ∖ 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).

Keywords

Chabauty topology, Heisenberg group, space of closed subgroups, space of lattices, affine group

Mathematical Subject Classification

Primary: 22D05, 22E25, 22E40

Milestones

Received: 18 December 2007
Accepted: 20 September 2008
Published: 2 March 2009

Authors