Geodesic
The transition constant for arithmetic hyperbolic reflection groups
Izvestiya. Mathematics , Tome 75 (2011) no. 5, pp. 971-1005.

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Using the results and methods of our papers [1], [2], we show that the degree of the ground field of an arithmetic hyperbolic reflection group does not exceed 25 in dimensions $n\geqslant 6$, and 44 in dimensions 3, 4, 5. This significantly improves our estimates obtained in [2]–[4]. We also use recent results in [5] and [6] to reduce the last bound to 35. We also review and correct the results of [1], § 1.
Keywords: group generated by reflections, arithmetic group, hyperbolic space, number field, field of definition, quadratic form. 
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V. V. Nikulin. The transition constant for arithmetic hyperbolic reflection groups. Izvestiya. Mathematics , Tome 75 (2011) no. 5, pp. 971-1005. http://geodesic.mathdoc.fr/item/IM2_2011_75_5_a5/

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