  Using authors's methods of 1980, 1981, some explicit finite sets of number
fields containing ground fields of arithmetic hyperbolic reflection groups are
defined, and good bounds of their degrees (over Q) are obtained. For example,
degree of the ground field of any arithmetic hyperbolic reflection group in
dimension at least 6 is bounded by 56. These results could be important for
further classification.
  We also formulate a mirror symmetric conjecture to finiteness of the number
of arithmetic hyperbolic reflection groups which was established in full
generality recently.
