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Discrete reflection groups in Lobachevsky spaces and algebraic surfaces. (English) Zbl 0671.22006

Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 654-671 (1987).
[For the entire collection see Zbl 0657.00005.]
This is the survey of some new results on discrete reflection groups in Lobachevsky spaces and of their applications to algebraic geometry: the author includes mainly the results which were not considered in E. B. Vinberg’s report at the 1983 International Congress of Mathematicians [1984; Zbl 0563.51011] devoted to the subject.
Section 2 is devoted to classification of crystallographic hyperbolic reflection groups (CHRG): the results of the author [Izv. Akad. Nauk SSSR, Ser. Mat. 44, 637-669 (1980; Zbl 0441.22008); ibid. 45, 113-142 (1981; Zbl 0477.22008)], A. G. Khovanskij [Linear programming and the geometry of convex polyhedra, Moskva, 73-81 (1985); Linear programming and a generalization of Nikulin’s theorem, Moskva, 81-87 (1985); Funkts. Anal. Prilozh. 20, 50-61 (1986; Zbl 0597.51014)], E. B. Vinberg [Mat. Sb., Nov. Ser. 72, 471-488 (1967; Zbl 0166.163); Funkts. Anal. Prilozh. 15, 67-68 (1981; Zbl 0462.51013); Tr. Mosk. Mat. O.-va. 47, 68-102 (1984; Zbl 0593.22007); Usp. Mat. Nauk 40, 29-66 (1985; Zbl 0579.51015)] and M. N. Prokhorov [Izv. Akad. Nauk SSSR, Ser. Mat. 50, 413-424 (1986; Zbl 0604.51007)], as well as some new method for bounding CHRG dimension are described.
Section 3 contains the description of two classes of groups to which, as the author believes, one can generalize the results of section 2.
Section 4 is devoted to applications of hyperbolic reflection groups to automorphisms of algebraic surfaces, namely, K3-surfaces, DPN-surfaces (one of the possible generalizations of Del Pezzo surfaces), Enriques surfaces (which are a special case of DPN-surfaces). The results, mainly obtained by the author [Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 18, 3-114 (1981; Zbl 0484.10021); Tr. Mat. Inst. Steklova 165, 119-142 (1984; Zbl 0577.10019); Dokl. Akad. Nauk SSSR 277, 1324-1327 (1984; Zbl 0604.14036)] are discussed.
Section 5 is devoted to hyperbolic reflection groups and real algebraic surfaces, mainly K3-surfaces. The discussion is based on the author’s results [published in Izv. Akad. Nauk SSSR, Ser. Mat. 43, 111-177 (1979; Zbl 0408.10011); ibid. 47, 109-188 (1983; Zbl 0547.10021); ibid. 49, 847- 873 (1985; Zbl 0585.10014); Itogi Nauki Tekh., Ser. Probl. Geom. 17, 87- 130 (1985; Zbl 0599.53029)].
Reviewer: V.L.Popov

MSC:

22E40 Discrete subgroups of Lie groups
20H15 Other geometric groups, including crystallographic groups
51F15 Reflection groups, reflection geometries
14J25 Special surfaces
14J28 \(K3\) surfaces and Enriques surfaces
51M10 Hyperbolic and elliptic geometries (general) and generalizations
11E57 Classical groups