  Following the previous work of Nikulin and Agol, Belolipetsky, Storm, and
Whyte it is known that there exist only finitely many (totally real) number
fields that can serve as fields of definition of arithmetic hyperbolic
reflection groups. We prove a new bound on the degree $n_k$ of these fields in
dimension 3: $n_k$ does not exceed 9. Combined with previous results of
Maclachlan and Nikulin, this leads to a new bound $n_k \le 25$ which is valid
for all dimensions. We also obtain upper bounds for the discriminants of these
fields and give some heuristic results which may be useful for the
classification of arithmetic hyperbolic reflection groups.
