Arithmetic hyperbolic reflection groups

Abstract

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.

Rigidity, lattices and invariant measures beyond homogeneous dynamics

“…The largest known non-arithmetic lattice produced by these methods is in dimension 18 by Vinberg and the full limits of reflection group constructions is not well understood [89]. We refer the reader to [11] for a detailed survey. The following question seems natural: Question 3.7.…”

The classification of rank 3 reflective hyperbolic lattices over

“…(2) In case (1), r k+2 = t, proceed to step (5). 36 (3) In case (2), compute h, the height of t. Do a restricted batch search of type I for roots of norm 2 whose mirrors intersect R k+1 in an angle of π 3 .…”

“…In two papers published at around the same time using different methods, Nikulin [15], and Agol, Belolipetsky, Storm and Whyte [2] both finished the proof of finiteness by proving it for dimensions 4 through 9. Belolipetsky's recent survey paper [5] gives a thorough overview the subject of hyperbolic reflection groups, including the history of this finitenesss theorem and many related results and problems.…”

On the classification of stably reflective hyperbolic Z[√2]-lattices of rank 4

“…[7] Наибольшее продвижение на данный момент достигнуто в классификации рефлективных гиперболических решёток над Z. Эти результаты хорошо освещены в недавнем обзоре М. Белолипецкого (см. [2]), а также в более новых работах автора данного сообщения, где классифицированы все устойчиво рефлективные анизотропные гиперболические Z -решётки ранга 4 (2016-2019, см. [3,4]).…”