Computing Arithmetic Subgroups of Affine Algebraic Groups
Andrea Pavan · Padua Research Archive (University of Padova) · 2009
This works deals with a problem concerning the algorithmic theory of affine algebraic groups. More precisely, it is possible to associate to any algebraic group defined over the field of the rational numbers a family of subgroups, the so-called arithmetic subgroups. In 1969, Borel and Harish-Chandra showed that every arithmetic group is finitely generated. Also, in the '80, Grunewald and Segal provided an algorithm for computing a finite set of generators of a given arithmetic subgroup of a given algebraic group. Unfortunately, their algorithm is not practical. In this work, we describe two original and practical algorithms for the same task, which work in the special cases in which the given algebraic group is unipotent or a torus, respectively.