Mahler measure under variations of the base group

We study properties of a generalization of the Mahler measure to elements in group rings, in terms of the Lueck-Fuglede-Kadison determinant. Our main focus is the variation of the Mahler measure when the base group is changed. In particular, we study how to obtain the Mahler measure over an infinite group as limit of Mahler measures over finite groups, for example, in the classical case of the free abelian group or the infinite dihedral group, and others.