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

Martin R. Bridson, Pierre de la Harpe and Victor Kleptsyn

Vol. 240 (2009), No. 1, 1–48

DOI: 10.2140/pjm.2009.240.1

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 2000

Primary: 22D05, 22E25, 22E40

Milestones

Received: 18 December 2007

Accepted: 20 September 2008

Published: 2 March 2009