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 $\Cal C(G)$. We analyse the structure of $\Cal C(G)$ for some low-dimensional Lie groups, concentrating mostly on the 3-dimensional Heisenberg group $H$. We prove that $\Cal C(H)$ is a 6-dimensional space that is path--connected but not locally connected. The lattices in $H$ form a dense open subset $\Cal L(H) \subset \Cal C(H)$ that is the disjoint union of an infinite sequence of pairwise--homeomorphic aspherical manifolds of dimension six, each a torus bundle over $({\mathbf{S}}^3\setminus T)\times \mathbf{R}$, where $T$ denotes a trefoil knot. The complement of $\Cal L(H)$ in $\Cal C(H)$ is also described explicitly. The subspace of $\Cal C(H)$ 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 $\Cal C(H)$.

Minor edits. Final version. To appear in the Pacific Journal. 41 pages, no figures

References in corpus (1)

• Limit groups as limits of free groups: compactifying the set of free groups

Cited by in corpus (1)

• Spaces of closed subgroups of locally compact groups