On the recurrence of coefficients in the L\"uck-Fuglede-Kadison determinant

TL;DR

This paper surveys how the Lück-Fuglede-Kadison determinant varies with different base groups and explores recurrence relations of its coefficients for specific groups, providing insights into its structural properties.

Contribution

It offers a comprehensive survey of the determinant's variations and introduces recurrence relations for coefficients in particular base groups, advancing understanding in this area.

Findings

Recurrences of coefficients identified for certain groups

Variations of the determinant with respect to base groups analyzed

Survey of existing results on the Lück-Fuglede-Kadison determinant

Abstract

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.