On ground fields of arithmetic hyperbolic reflection groups. II

TL;DR

This paper extends previous work to explicitly identify finite sets of number fields containing all ground fields of arithmetic hyperbolic reflection groups in dimensions ≥4, providing bounds on their degrees, which aids in classification efforts.

Contribution

It defines explicit finite sets of number fields containing all ground fields for dimensions ≥4 and establishes bounds on their degrees, advancing the classification of these groups.

Findings

Finite sets of number fields are explicitly constructed.

Bounds on degrees of ground fields are obtained.

Classification of hyperbolic reflection groups is facilitated.

Abstract

This paper continues arXiv.org:math.AG/0609256 and arXiv:0708.3991 Using authors's methods of 1980, 1981, some explicit finite sets of number fields containing all ground fields of arithmetic hyperbolic reflection groups in dimensions at least 4 are defined, and good explicit bounds of their degrees (over Q) are obtained. This could be important for further classification. Thus, now, an explicit bound of degree of ground fields of arithmetic hyperbolic reflection groups is unknown in dimension 3 only.