Fuglede-Kadison determinants over free groups and Lehmer's constants
(2022) Confluentes Mathematici — Vol. 14, n° 1, p. 3-22 (2022)
Files
BA6_CML_2022__14_1_3_0.pdf
Published version
Open Access - Adobe PDF
- 726.46 KB
Details
- Authors
- 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 2/√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 L2-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 2/√3.
- Affiliations
Citations
Ben Aribi, F. (2022). Fuglede-Kadison determinants over free groups and Lehmer’s constants. Confluentes Mathematici, 14(1), 3-22. https://doi.org/10.5802/cml.79 (Original work published 2022)