On methods of classification of reflective hyperbolic lattices

A hyperbolic lattice is a free Z-module of finite rank with an inner product of signature (n,1). A hyperbolic lattice L is said to be reflective if the subgroup R(L) of its automorphism group generated by all reflections is of finite index. It is equivalent to the fact that the fundamental polyhedron of the group R(L) is the n-dimensional hyperbolic Coxeter polyhedron of finite volume.
Classification of reflective hyperbolic lattices is an old open problem posed in 1967 by E.B. Vinberg. V.V. Nikulin (1980, 2007) has proved that there are only finitely many of reflective hyperbolic lattices in all dimensions, and the dimensions, for which reflective lattices exist, were bounded by E.B. Vinberg and F. Esselman (1984, 1996).
At present, reflective lattices of rank 3 are classified (V.V. Nikulin, 2000, and finally, D. Allcock, 2011), as well as isotropic reflective lattices of rank 4 (R.Sharlau, 1989). The reflective lattices of rank 5 (R.Sharlau, K. Walhorn, 1992-1993) and rank 6 (I. Turkal, 2017) are also classified. Refletive anisotropic hyperbolic lattices of rank 4, as well as reflective lattices of ranks above 6, have not yet been classified.
I will describe in the talk a new method of classification of reflective hyperbolic lattices (which is a modification of the method applied by V.V. Nikulin), which I managed to apply for the classification of (1.2)-reflective anisotropic hyperbolic lattices of rank 4 (that is, for lattices whose automorphism groups is generated by 1- and 2-reflections up to finite index). I will also talk about the computer implementation of Vinberg's Algorithm (joint with A.Yu. Perepechko) constructing of a fundamental polyhedron for groups of type R(L).