  In this note, we survey results concerning variations of the
L\"uck-Fuglede-Kadison determinant with respect to the base group. Further, we
discuss recurrences of coefficients in the determinant for certain
distinguished base groups. The note is based on a talk that the second author
gave at the "Segundas Jornadas de Teor\'ia de N\'umeros", Madrid, 2007.
