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Limits of Lattices in a Compactly Generated Group

  • A. M. Macbeath (a1) and S. Świerczkowski (a2)
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Let G be a locally compact and (σ-compact topological group and let H be a discrete subgroup of G. We shall use G/H to denote the space of right cosets Hx of H with the usual topology (cf. (8, pp. 26-28)). Let μ be the left Haar measure in G. μ induces a measure in the space G/H3; this measure will, without ambiguity in this paper, also be denoted by μ. If μ(G/H) is finite, the group H is called a lattice. If the space G/H is compact, then H is certainly a lattice and is called a bounded lattice. These terms are an extension of the usage of the Geometry of Numbers, where G is the real n-dimensional vector space Rn . In this case any lattice is generated by n linearly independent vectors, all lattices are bounded, and the whole family of lattices is permuted transitively by the automorphisms of G (which are the non-singular linear transformations).

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References
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1. Chabauty, C., Limite d'ensembles et géométrie des nombres, Bull. Soc. Math. France, 78 (1950), 143151.
2. Ford, L.R., Automorphic functions (New York: Chelsea, 1951).
3. Fricke, R. and Klein, F., Vorlesungen ueber die Théorie der Automorphen Funktionen (Leipzig: Teubner, 1897-1912).
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5. Macbeath, A.M., Abstract theory of packings and coverings, I (to appear in Proc. Glasgow Math. Assoc).
6. Macbeath, A.M. and Swierczkowski, S., On the set of generators of a subgroup, Indag. Math., 21 (1959), 280281.
7. Mahler, K., On lattice points in n-dimensional star bodies. I, Existence Theorems, Proc. Roy. Soc. London, Ser. A.187 (1946), 151187.
8. Montgomery, D. and Zippin, L., Topological transformation groups (New York: Interscience tracts, 1955).
9. Siegel, C.L., Discontinuous groups, Ann. Math., 44 (1943), 674678.
10. Swierczkowski, S., Abstract theory of packings and coverings, II (to appear in Proc. Glasgow Math. Assoc).
11. Weil, A., Vintegration dans les groupes topologiques et ses applications (Paris: Hermann, 1951).
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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