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On Lie groups with lattices and their properties. (English. Russian original) Zbl 0619.22014

Sov. Math., Dokl. 33, 321-325 (1986); translation from Dokl. Akad. Nauk SSSR 287, 33-37 (1986).
Some results about lattices in Lie groups are announced. (As usual, by a lattice \(\Gamma\) in a Lie group G we mean a discrete subgroup such that the volume of the quotient space G/\(\Gamma\) is finite.) The main result is the following
Theorem 1. The set of all connected Lie groups, considered up to isomorphism and containing uniform lattices, is countable. (The lattice \(\Gamma\) in G is called uniform if the space G/\(\Gamma\) is compact.)
Let us mention two other results announced in the paper: Theorem 2. The number of connected semisimple Lie groups G (considered up to isomorphism) without compact factors containing uniform lattices isomorphic to a given group is always finite.
Theorem 9. Let M be a compact homogeneous manifold. Then Novikov’s conjecture on the homotopy invariance of higher signatures is valid for its fundamental group \(\pi_ 1(M).\)
It is written in the paper that theorem 9 is an application of the results about lattices in Lie groups.
Reviewer: G.A.Margulis

MSC:

22E40 Discrete subgroups of Lie groups
22E46 Semisimple Lie groups and their representations
57R20 Characteristic classes and numbers in differential topology
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