Discrete subgroups of Lie groups.

*(English)*Zbl 0605.22008
Élie Cartan et les mathématiques d’aujourd’hui. The mathematical heritage of Élie Cartan, Semin. Lyon 1984, Astérisque, No.Hors. Sér. 1985, 289-309 (1985).

[For the entire collection see Zbl 0573.00010.]

The reviewed paper is a survey on the theory of lattice subgroups, i.e. discrete subgroups \(\Gamma\) of a Lie group G with G/\(\Gamma\) having finite Haar measure. In the introduction, the historical background of the theory is described. In the next two sections, the definition of an arithmetic subgroup and the theorem on arithmeticity of irreducible lattices in R-rank \(>1\) semisimple Lie groups are given.

The remainder of the paper is devoted to results on construction of lattices by methods other than the arithmetic definition, namely (1) a construction of subgroups in PO(n,1) generated by reflections; (2) a construction of subgroups in PU(n,1) generated by complex reflections; (3) a construction on lattices in PU(n,1) via monodromy. A construction of negatively curved surfaces not covered by the ball is also given. The following two conjectures are formulated in the last section.

Conjecture 1. Except in low dimensions, lattices in R-rank 1 groups are arithmetic.

Conjecture 2. Except in low dimensions, there are no lattices in PU(n,1) generated by complex reflections.

It should be noted that Gromov and Piatetskij-Shapiro recently gave examples of nonarithmetic lattices in PO(n,1) for arbitrary n. Thus the answer to conjecture 1 is negative.

The reviewed paper is a survey on the theory of lattice subgroups, i.e. discrete subgroups \(\Gamma\) of a Lie group G with G/\(\Gamma\) having finite Haar measure. In the introduction, the historical background of the theory is described. In the next two sections, the definition of an arithmetic subgroup and the theorem on arithmeticity of irreducible lattices in R-rank \(>1\) semisimple Lie groups are given.

The remainder of the paper is devoted to results on construction of lattices by methods other than the arithmetic definition, namely (1) a construction of subgroups in PO(n,1) generated by reflections; (2) a construction of subgroups in PU(n,1) generated by complex reflections; (3) a construction on lattices in PU(n,1) via monodromy. A construction of negatively curved surfaces not covered by the ball is also given. The following two conjectures are formulated in the last section.

Conjecture 1. Except in low dimensions, lattices in R-rank 1 groups are arithmetic.

Conjecture 2. Except in low dimensions, there are no lattices in PU(n,1) generated by complex reflections.

It should be noted that Gromov and Piatetskij-Shapiro recently gave examples of nonarithmetic lattices in PO(n,1) for arbitrary n. Thus the answer to conjecture 1 is negative.

Reviewer: G.A.Margulis