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

Martin R Bridson, Pierre de la Harpe, and Victor Klepstyn
MSC: 22D05, 22E25; 22E40

To appear in the Pacific Journal of Mathematics

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 $(\bold S^3 \smallsetminus T) \times \bold R$,
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 $\Cal C(H)$.