Effective bounds for Vinberg’s algorithm for arithmetic hyperbolic lattices

A group of isometries a hyperbolic n-space is called reflection if it generated by reflections in hyperplanes. Vinberg gave semi-algorithm for finding maximal sublattice given arithmetic subgroup $${{\,\textrm{O}\,}}(n,1)$$ the simplest type. We provide an effective termination condition Vinberg’s with which becomes algorithm sublattices. The main new ingredient proof upper bound number faces Coxeter polyhedron terms its volume.