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

After results of the author (1980, 1981) and Vinberg (1981), the finiteness of the number of maximal arithmetic groups generated by reflections in Lobachevsky spaces remained unknown in dimensions 2≤n≤9 only. It was proved recently (2005) in dimension 2 by Long, Maclachlan and Reid and in dimension 3 by Agol. Here we use the results in dimensions 2 and 3 to prove the finiteness in all remaining dimensions 4≤n≤9. The methods of the author (1980, 1981) are more than sufficient for this using a very short and very simple argument