### Speaker

Nikolay Bogachev
(Moscow Institute of Physics and Technology)

### Description

By definition, a hyperbolic lattice is a free Abelian group with an integral inner product of signature $(n,1)$.
A hyperbolic lattice is said to be reflective if the subgroup $O_r (L)$ of its automorphism group generated
by all reflections is of finite index. The lattice $L$ is reflective if and only if the fundamental polyhedron $P$
of the group $O_r (L)$ has a finite volume in the Lobachevsky (hyperbolic) space $\mathbb{L}^n$.
There is Vinberg's Algorithm that, given a lattice $L$, enables one to find recursively all faces of the
polyhedron $P$ and determine if there are only finitely many of them. In particular, it enables one to
test a given lattice $L$ for reflectivity.
In this talk I present some new methods of classification of arithmetic hyperbolic reflection groups and
new results on classification of reflective hyperbolic lattices obtained by these methods. In addition,
I present a software implementation of Vinberg's Algorithm for hyperbolic lattices (joint work with Alexander Perepechko).