SOLVABLE ARITHMETIC GROUPS AND ARITHMETICITY PROBLEMS

Abstract

We describe solvable arithmetic groups for which natural rigidity properties hold and solve the arithmeticity problem for the automorphism groups of these arithmetic groups and further prove arithmeticity results for their finite extensions. We also solve the arithmeticity problem for polycyclic groups. We prove that there are non-arithmetic polycyclic groups of every Hirsch number >3 and that all polycyclic groups of Hirsch number ≤3 are arithmetic. In fact we work with the more general class of finite extensions of solvable arithmetic groups and obtain results in this case.