BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 53, Number 3, July 2016, Pages 437–475 http://dx.doi.org/10.1090/bull/1530 Article electronically published on March 29, 2016
ARITHMETIC HYPERBOLIC REFLECTION GROUPS
MIKHAIL BELOLIPETSKY Dedicated to Ernest Borisovich Vinberg
Abstract. A hyperbolic reflection group is a discrete group generated by reflections in the faces of an n-dimensional hyperbolic polyhedron. This survey article is dedicated to the study of arithmetic hyperbolic reflection groups with an emphasis on the results that were obtained in the last ten years and on the open problems.
Contents 1. Introduction 437 2. Quadratic forms and arithmetic reflection groups 440 3. The work of Nikulin 445 4. Spectral method and finiteness theorems 447 5. Effective results obtained by the spectral method 449 6. Classification results in small dimensions 452 7. Examples 456 8. Reflective modular forms 459 9. More about the structure of the reflective quotient 462 10. Open problems 464 Acknowledgments 468 About the author 468 References 468
1. Introduction Consider a finite volume polyhedron P in the n-dimensional hyperbolic space Hn. It may occur that if we act on P by the hyperbolic reflections in its sides, the images would cover the whole space Hn and would not overlap with each other. In this case we say that the transformations form a hyperbolic reflection group Γ and that P is its fundamental polyhedron, also known as the Coxeter polyhedron of Γ. We can give an analogous definition of the spherical and euclidean reflection groups, the classes of which are well understood after the work of Coxeter [Cox34]. What kind of properties characterize hyperbolic Coxeter polyhedra? For example, π we have to assume that all the dihedral angles of P are of the form m for some m ∈{2, 3,...,∞} because otherwise some images of P over Γ would overlap. It
Received by the editors September 21, 2015. 2010 Mathematics Subject Classification. Primary 22E40; Secondary 11F06, 11H56, 20H15, 51F15.