Mahler measures and Fuglede--Kadison determinants

Abstract

The Mahler measure of a function on the real d-torus is its geometric mean over the torus. It appears in number theory, ergodic theory and other fields. The Fuglede-Kadison determinant is defined in the context of von Neumann algebra theory and can be seen as a noncommutative generalization of the Mahler measure. In the paper we discuss and compare theorems in both fields, especially approximation theorems by finite dimensional determinants. We also explain how to view Fuglede-Kadison determinants as continuous functions on the space of marked groups.

References in corpus (4)

• L2-determinant class and approximation of L2-Betti numbers
• Limit groups as limits of free groups: compactifying the set of free groups
• Mahler measures and computations with regulators
• Determinants on von Neumann algebras, Mahler measures and Ljapunov exponents

Cited by in corpus (8)

• Homoclinic groups, IE groups, and expansive algebraic actions
• Survey on approximating L^2-invariants by their classical counterparts: Betti numbers, torsion invariants and homological growth
• Compact Group Automorphisms, Addition Formulas and Fuglede-Kadison Determinants
• Entropy, Determinants, and L2-Torsion
• Survey on L^2-invariants and 3-manifolds
• Combinatorial independence and sofic entropy
• Limits of Mahler measures in multiple variables
• Independence Tuples and Deninger's Problem