The Chabauty space of closed subgroups
To appear in the Pacific Journal of Mathematics
of the three-dimensional Heisenberg group
Martin R Bridson, Pierre de la Harpe, and Victor Klepstyn
MSC: 22D05, 22E25; 22E40
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 $\Cal C(H)$.