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

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dc.creatorBridson, Martin R.
dc.creatorde la Harpe, Pierre
dc.creatorKleptsyn, Victor
dc.date2007-11-23
dc.date2008-11-18
dc.date.accessioned2026-07-07T10:18:31Z
dc.date.available2026-07-07T10:18:31Z
dc.descriptionWhen 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 $\Cal C(H)$.
dc.descriptionMinor edits. Final version. To appear in the Pacific Journal. 41 pages, no figures
dc.identifierhttps://arxiv.org/abs/0711.3736
dc.identifierhttp://arxiv.org/abs/0711.3736
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/174213
dc.subjectGroup Theory
dc.subject22D05, 22E25, 22E40
dc.titleThe Chabauty space of closed subgroups of the three-dimensional Heisenberg group
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