Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-25T22:58:14.833Z Has data issue: false hasContentIssue false
  • English
  • Français

Limits of Lattices in a Compactly Generated Group

Published online by Cambridge University Press:  20 November 2018

A. M. Macbeath  and
S. Świerczkowski
A. M. Macbeath
Affiliation:
Queen's College
S. Świerczkowski
Affiliation:
Dundee, Scotland
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 12 , 1960 , pp. 427 - 437
Copyright
Copyright © Canadian Mathematical Society 1960

References

1. Chabauty, C., Limite d'ensembles et géométrie des nombres, Bull. Soc. Math. France, 78 (1950), 143–151.Google Scholar
2. Ford, L.R., Automorphic functions (New York: Chelsea, 1951).Google Scholar
3. Fricke, R. and Klein, F., Vorlesungen ueber die Théorie der Automorphen Funktionen (Leipzig: Teubner, 1897-1912).Google Scholar
4. Halmos, P.R., Measure theory (New York: Van Nostrand, 1950).Google Scholar
5. Macbeath, A.M., Abstract theory of packings and coverings, I (to appear in Proc. Glasgow Math. Assoc).Google Scholar
6. Macbeath, A.M. and Swierczkowski, S., On the set of generators of a subgroup, Indag. Math., 21 (1959), 280–281.Google Scholar
7. Mahler, K., On lattice points in n-dimensional star bodies. I, Existence Theorems, Proc. Roy. Soc. London, Ser. A.187 (1946), 151–187.Google Scholar
8. Montgomery, D. and Zippin, L., Topological transformation groups (New York: Interscience tracts, 1955).Google Scholar
9. Siegel, C.L., Discontinuous groups, Ann. Math., 44 (1943), 674–678.Google Scholar
10. Swierczkowski, S., Abstract theory of packings and coverings, II (to appear in Proc. Glasgow Math. Assoc).Google Scholar
11. Weil, A., Vintegration dans les groupes topologiques et ses applications (Paris: Hermann, 1951).Google Scholar