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

Fathi Ben Aribi

doi:10.5802/cml.79

Fathi Ben Aribi ¹

¹ UCLouvain, IRMP, Chemin du Cyclotron 2; 1348 Louvain-la-Neuve; Belgium

Confluentes Mathematici, Tome 14 (2022) no. 1, pp. 3-22.

Résumé

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

Reçu le : 2022-01-27
Révisé le : 2022-08-18
Accepté le : 2022-08-18
Publié le : 2022-11-21

DOI : 10.5802/cml.79

Classification : 57K10, 57M05, 20F36, 11R06, 47C15
Mots clés : $L^2$-invariants; braid groups; Fuglede-Kadison determinant; Lehmer’s constants

Affiliations des auteurs :

Fathi Ben Aribi ¹

¹ UCLouvain, IRMP, Chemin du Cyclotron 2; 1348 Louvain-la-Neuve; Belgium

Licence :

CC-BY-NC-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Fathi Ben Aribi. Fuglede-Kadison determinants over free groups and Lehmer’s constants. Confluentes Mathematici, Tome 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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