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Fuglede-Kadison determinants over free groups and Lehmer’s constants
Fathi Ben Aribi1
1 UCLouvain, IRMP, Chemin du Cyclotron 2; 1348 Louvain-la-Neuve; Belgium
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 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 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 2 3.

Received:
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
License: CC-BY-NC-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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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/

[1] M. Aschenbrenner, S. Friedl and H. Wilton. 3-manifold groups, EMS Series of Lectures in Mathematics. European Mathematical Society (EMS), xiv+215 pp., Zürich, 2015. | DOI | MR | Zbl

[2] L. Bartholdi. Counting paths in graphs, Enseign. Math. (2) 45, no. 1-2, 83–131, 1999. | Zbl

[3] F. Ben Aribi. A study of properties and computation techniques of the L 2 -Alexander invariant in knot theory, PhD thesis, Université Paris Diderot, Paris, 2015.

[4] F. Ben Aribi. Gluing formulas for the L 2 -Alexander torsions, Commun. Contemp. Math. 21, no. 3, 1850013, 31 pp., 2019. | DOI | Zbl

[5] F. Ben Aribi and A. Conway. L 2 -Burau maps and L 2 -Alexander torsions, Osaka J. Math. 55, 529–545, 2018. | Zbl

[6] D. Boyd. Speculations concerning the range of Mahler’s measure, Canad. Math. Bull. 24, no. 4, 453–469, 1981. | DOI | MR | Zbl

[7] M. R. Bridson, D. B. McReynolds, A. W. Reid, R. Spitler. Absolute profinite rigidity and hyperbolic geometry, Ann. of Math. (2) 192, no. 3, 679–719, 2020. | DOI | MR | Zbl

[8] W. Burau. Über Zopfgruppen und gleichsinnig verdrillte Verkettungen, Abh. Math. Sem. Univ. Hamburg 11, 179–186, 1935. | DOI | Zbl

[9] Marc Culler, Nathan M. Dunfield, Matthias Goerner, and Jeffrey R. Weeks. SnapPy, a computer program for studying the geometry and topology of 3-manifolds, available at http://snappy. computop.org

[10] O. Dasbach and M. Lalin. Mahler measure under variations of the base group, Forum Math. 21, no. 4, 621–637, 2009. | DOI | MR | Zbl

[11] J. Dubois, S. Friedl and W. Lück. The L 2 -Alexander torsion of 3-manifolds, Journal of Topology 9, No. 3, 889–926, 2016. | DOI | Zbl

[12] R.H. Fox. Free differential calculus. II. The isomorphism problem of groups, Ann. of Math. (2), 59, 196–210, 1954. | DOI | MR | Zbl

[13] D. Gabai, R. Meyerhoff and P. Milley. Minimum volume cusped hyperbolic three-manifolds, J. Amer. Math. Soc. 22(4), 1157–1215, 2009. | DOI | MR | Zbl

[14] A. Kricker and Z. Wong. Random Walks on Graphs and Approximation of L 2 -Invariants, Acta Mathematica Vietnamica, Volume 46, Issue 2, pp. 309–319, March 2021. | DOI | Zbl

[15] W. Li and W. Zhang. An L 2 -Alexander invariant for knots, Commun. Contemp. Math. 8, no. 2, 167–187, 2006. | DOI | Zbl

[16] Y. Liu. Degree of L 2 -Alexander torsion for 3-manifolds, Invent. Math. 207, 981–1030, 2017. | DOI | MR | Zbl

[17] W. Lück. L 2 -invariants: theory and applications to geometry and K-theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 44. Springer-Verlag, Berlin, 2002. | Zbl

[18] W. Lück. Lehmer’s Problem for arbitrary groups, to appear in Journal of Topology and Analysis, arXiv:1901.00827, 2019. | DOI

[19] W. Lück and T. Schick. L 2 -torsion of hyperbolic manifolds of finite volume, Geom. Funct. Anal. 9, no. 3, 518–567, 1999. | DOI | MR | Zbl

[20] Wolfram Research, Inc., Wolfram|Alpha Notebook Edition, Champaign, IL (2022), https://www.wolframalpha.com/.

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