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arXiv · 2202.03877

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

Fathi Ben Aribi
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Abstract

Lehmer's famous problem asks whether the set of Mahler measures of polynomials with integer coefficients admits a gap at 1. In 2019, Lück 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 $\frac{2}{\sqrt{3}}$.

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Fathi Ben Aribi. 2022-08-31. Fuglede-Kadison determinants over free groups and Lehmer's constants. https://arxiv.org/abs/2202.03877

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