# CONFLUENTES MATHEMATICI

Fuglede-Kadison determinants over free groups and Lehmer’s constants
Confluentes Mathematici, Volume 14 (2022) no. 1, pp. 3-22.

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 $\frac{2}{\sqrt{3}}$.

Revised:
Accepted:
Published online:
DOI: 10.5802/cml.79
Classification: 57K10,  57M05,  20F36,  11R06,  47C15
Keywords: ${L}^{2}$-invariants; braid groups; Fuglede-Kadison determinant; Lehmer’s constants
Fathi Ben Aribi 1

1 UCLouvain, IRMP, Chemin du Cyclotron 2; 1348 Louvain-la-Neuve; Belgium
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Fathi Ben Aribi. Fuglede-Kadison determinants over free groups and Lehmer’s constants. Confluentes Mathematici, Volume 14 (2022) no. 1, pp. 3-22. doi : 10.5802/cml.79. https://cml.centre-mersenne.org/articles/10.5802/cml.79/

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