Spaces of Closed Subgroups of Locally Compact Groups
نویسنده
چکیده
The set C(G) of closed subgroups of a locally compact group G has a natural topology which makes it a compact space. This topology has been defined in various contexts by Vietoris, Chabauty, Fell, Thurston, Gromov, Grigorchuk, and many others. The purpose of the talk was to describe the space C(G) first for a few elementary examples, then for G the complex plane, in which case C(G) is a 4–sphere (a result of Hubbard and Pourezza), and finally for the 3–dimensional Heisenberg group H, in which case C(H) is a 6–dimensional singular space recently investigated by Martin Bridson, Victor Kleptsyn and the author [BrHK]. These are slightly expanded notes prepared for a talk given at several places: the Kortrijk workshop on Discrete Groups and Geometric Structures, with Applications III, May 26–30, 2008; the Tripode 14, École Normale Supérieure de Lyon, June 13, 2008; and seminars at the EPFL, Lausanne, and in the Université de Rennes 1. The notes do not contain any other result than those in [BrHK], and are not intended for publication. I. Mahler (1946) Let n be a positive integer. A lattice in R is a subgroup of R generated by a basis. Two lattices L,L ⊂ R ar close to each other if there exist basis {e1, . . . , en} ⊂ L, {e1, . . . , en} ⊂ L with ej and ej close to each other for each j; this defines a topology on the space L(Rn) of lattices in R. It coincides with the natural topology on L(Rn) viewed as the homogeneous space GLn(R)/GLn(Z). It is easy to check that the covolume function L 7−→ vol(R/L) and the minimal norm L 7−→ minL + minx∈L,x6=0 ‖x‖2 are continuous functions L(Rn) −→ R+. 1. Mahler’s Criterion [Mahl–46]. For a subset M of L(Rn), the two following properties are equivalent: (i) M is relatively compact; (ii) there exist C, c > 0 such that vol(R/L) ≤ C and minL ≥ c for all L ∈ M. For proofs of this criterion, see for example [Bore–69, corollaire 1.9] or [Ragh–72, Corollary 10.9]. An immediate use of the criterion is the proof that, in any dimension n ≥ 1, the space Lunimod(Rn) of lattices of covolume 1 contains a lattice Lmax such that minLmax = sup{minL | L ∈ Lunimod(Rn)}; equivalently, there exists a lattice Lmax with a maximal 2000 Mathematics Subject Classification. 22D05, 22E25, 22E40.
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