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Hyperbolic reflection groups



Reflection groups are beautiful groups which arise naturally as symmetry groups of various mathematical objects as well as everyday life objects and art pieces. By a reflection group we mean a group generated by reflections with respect to some hyperplanes (or with respect to mirrors). The most interesting ones are the reflection groups acting discretely on a (symmetric) space, that means that the images of the mirrors never accumulate around any points of the space. This type of groups is a point where the interests of a child watching a kaleidoscope meet the interests of a scientist studying Riemmanian manifolds, or automorphic forms, or hypergeometric functions, or orbifolds, or arithmetics of quadratic forms, or K3 surfaces, etc.

In this minicourse we will use regular polytopes to discuss general properties of reflection groups and Coxeter groups. We will consider discrete reflection groups in spherical and Euclidean spaces, and then we will focus on reflection groups acting discretely on a hyperbolic space. This space is really rich by examples of infinite reflection groups. On the other hand, hyperbolic reflection groups are far from being classified. We will show some classical results concerning hyperbolic reflection groups as well as describe some recent progress in this field.

No special knowledge is required!