Finiteness of the number of arithmetic groups generated by reflections in Lobachevsky spaces

Abstract

After results by the author (1980, 1981), and by Vinberg (1981), finiteness of the number of maximal arithmetic reflection groups in Lobachevsky spaces was not known in dimensions $2\le n\le 9$ only. Recently (2005), the finiteness was proved in dimension 2 by Long, Maclachlan and Reid, and in dimension 3 by Agol. Here we use these results in dimensions 2 and 3 to prove finiteness in all remaining dimensions $4\le n\le 9$. Methods of the author (1980, 1981) are strong enough to complete this in few lines by simple considerations.