Hyperbolic reflection groups. (English. Russian original) Zbl 0579.51015

Russ. Math. Surv. 40, No. 1, 31-75 (1985); translation from Usp. Mat. Nauk 40, No. 1(241), 29-66 (1985).
An abstract group \(\Gamma\) with a finite set of generators \(R_i\) is called the Coxeter group if \(R^2_i=1\), \((R_i\cdot R_j)^{n_{ij}}=1\), where \(n_{ij}\geq 2\). J. Tits [Symp. Math. 1, 175–185 (1969; Zbl 0206.03002)] proved that every Coxeter group with the finite set of generators is represented by the reflexive group discrete in some domain of the projective space. All representations of this type are described by the author [see Math. USSR, Izv. 5, 1083–1119 (1971); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 35, 1072–1112 (1971; Zbl 0247.20054)].
The object of this paper is to investigate the properties of hyperbolic Coxeter group whose representations are discrete reflection groups in the Lobachevsky space.


51F15 Reflection groups, reflection geometries
20H15 Other geometric groups, including crystallographic groups
51M10 Hyperbolic and elliptic geometries (general) and generalizations
52Bxx Polytopes and polyhedra
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