Fuglede-Kadison determinants over free groups and Lehmer's constants

TL;DR

This paper computes new bounds for Fuglede-Kadison determinants over free groups, leading to improved upper bounds for Lehmer's constants in certain torsion-free and hyperbolic 3-manifold groups, connecting algebraic and geometric group properties.

Contribution

It provides new explicit values for Fuglede-Kadison determinants over free groups and establishes tighter bounds for Lehmer's constants in specific classes of groups.

Findings

New upper bound $rac{2}{rac{ ext{sqrt}(3)}}$ for Lehmer's constants in non-cyclic free groups.

Bounded Lehmer's constants for fundamental groups of hyperbolic 3-manifolds.

Relations between Fuglede-Kadison determinants and random walks on Cayley graphs.

Abstract

Lehmer's famous problem asks whether the set of Mahler measures of polynomials with integer coefficients admits a gap at 1. In 2019, L\"uck extended this question to Fuglede-Kadison determinants of a general group, and he defined the Lehmer's constants of the group to measure such a gap. In this paper, we compute new values for Fuglede-Kadison determinants over non-cyclic free groups, which yields the new upper bound $\frac{2}{\sqrt{3}}$ for Lehmer's constants of all torsion-free groups which have non-cyclic free subgroups. Our proofs use relations between Fuglede-Kadison determinants and random walks on Cayley graphs, as well as works of Bartholdi and Dasbach-Lalin. Furthermore, via the gluing formula for $L^2$-torsions, we show that the Lehmer's constants of an infinite number of fundamental groups of hyperbolic 3-manifolds are bounded above by even smaller values than $\frac{2}{\sqrt{3}}$.