Arithmetic hyperbolic reflection groups (Q2810914)

A hyperbolic reflection group is a discrete group generated by reflections in the faces of an \(n\)-dimensional hyperbolic polyhedron.NEWLINENEWLINEThe author discusses many open problems concerning hyperbolic reflection groups. He deals primarily with recent results. For earlier results he refers to a survey of [\textit{Eh. B. Vinberg}, Russ. Math. Surv. 40, No. 1, 31--75 (1985; Zbl 0579.51015); translation from Usp. Mat. Nauk 40, No. 1(241), 29--66 (1985)].NEWLINENEWLINEThe author gives a lively and comprehensive account on hyperbolic reflection groups. He discusses many open problems. His starting point is the following result of Vinberg: `There are no arithmetic hyperbolic reflection groups in spaces with dimensions \(n \geq 30\)'. He asks for which dimensions do hyperbolic reflection groups exist. He presents a detailed discussion about the existence of arithmetic hyperbolic reflection groups. He conjectures that similar results can be obtained if the assumption `arithmetic' is omitted. The author discusses extensively what happens in spaces with dimensions for which arithmetic reflection groups exist. How can examples of such groups be constructed and is it possible to classify all of them?NEWLINENEWLINEMany other topics are included in this survey, such as a question of Anton Petrunin: `Do there exist hyperbolic lattices (i.e., discrete cofinite isometry groups) generated by elements of finite order in spaces of large dimension?' It is expected that the answer is `no'.