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

Fathi Ben Aribi

Published 2022-02-08Version 1

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. As a consequence, we partially answer L\"uck's question, as we provide the new upper bound $\frac{2}{\sqrt{3}}$ for Lehmer's constants of all torsionfree 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, we study several fundamental groups of small hyperbolic 3-manifolds, and we show that all of their Lehmer's constants are bounded by even smaller values than $\frac{2}{\sqrt{3}}$. For this, we use relations between Fuglede-Kadison determinants, $L^2$-torsions and hyperbolic volumes. Finally, we describe a method for computing upper approximations of Fuglede-Kadison determinants and we apply it for the group of the figure-eight knot complement.