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

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).Comment: Minor edits. Final version. To appear in the Pacific Journal. 41 pages, no figure

Kneser-Hecke-operators in coding theory

The Kneser-Hecke-operator is a linear operator defined on the complex vector space spanned by the equivalence classes of a family of self-dual codes of fixed length. It maps a linear self-dual code $C$ over a finite field to the formal sum of the equivalence classes of those self-dual codes that intersect $C$ in a codimension 1 subspace. The eigenspaces of this self-adjoint linear operator may be described in terms of a coding-theory analogue of the Siegel $\Phi$-operator