The transition constant for arithmetic hyperbolic reflection groups

TL;DR

This paper refines the upper bound of the transition constant for arithmetic hyperbolic reflection groups, which is crucial for classifying these groups based on their ground field degrees.

Contribution

It improves the known upper bound of the transition constant from 56 to a lower value, aiding in the classification of arithmetic hyperbolic reflection groups.

Findings

The upper bound of the transition constant is improved.

Results are relevant for classifying hyperbolic reflection groups.

The paper advances understanding of ground field degrees in these groups.

Abstract

The transition constant was introduced in our 1981 paper and denoted as N(14). It is equal to the maximal degree of the ground fields of V-arithmetic connected edge graphs with 4 vertices and of the minimality 14. This constant is fundamental since if the degree of the ground field of an arithmetic hyperbolic reflection group is greater than N(14), then the field comes from very special plane reflection groups. In our recent paper (see also arXiv:0708.3991), we claimed its upper bound 56. Using similar but more difficult considerations, here we improve this bound. These results could be important for further classification.