Arithmetic Hyperbolic Reflection Groups

Abstract

This thesis uses Vinberg’s algorithm to study arithmetic hyperbolic reflection groups which are contained in the groups of units of quadratic forms. We study two families
of quadratic forms: the diagonal forms −dx_0^2 + x_1^2 + ... + x_n^2 ; and the forms whose automorphism groups contain the Bianchi groups. In the first instance we classify over Q the pairs (d,n) for which such a group can be found, and in some cases we can compute the volumes of the fundamental polytopes.
In the second instance we use a combination of the geometric and number theoretic information to classify the reflective Bianchi groups by first classifying the reflective extended Bianchi groups, namely the maximal discrete extension of the Bianchi groups in PSL(2,C).
Finally we identify some quadratic forms in the first instance and completely classify those in the second which have a quasi-reflective structure.