Skip to main content

$p$ -adic Mahler measure and $\mathbb{Z}$ -covers of links

  • JUN UEKI (a1)

Let $p$ be a prime number. We develop a theory of $p$ -adic Mahler measure of polynomials and apply it to the study of $\mathbb{Z}$ -covers of rational homology 3-spheres branched over links. We obtain a $p$ -adic analogue of the asymptotic formula of the torsion homology growth and a balance formula among the leading coefficient of the Alexander polynomial, the $p$ -adic entropy and the Iwasawa $\unicode[STIX]{x1D707}_{p}$ -invariant. We also apply the purely $p$ -adic theory of Besser–Deninger to $\mathbb{Z}$ -covers of links. In addition, we study the entropies of profinite cyclic covers of links. We examine various examples throughout the paper.

Hide All
[AKM65] Adler, R. L., Konheim, A. G. and McAndrew, M. H.. Topological entropy. Trans. Amer. Math. Soc. 114 (1965), 309319.
[BD99] Besser, A. and Deninger, C.. p-adic Mahler measures. J. Reine Angew. Math. 517 (1999), 1950.
[Bow71] Bowen, R.. Entropy for group endomorphisms and homogeneous spaces. Trans. Amer. Math. Soc. 153 (1971), 401414.
[Boy02] Boyd, D. W.. Mahler’s measure and invariants of hyperbolic manifolds. Number Theory for the Millennium, I (Urbana, IL, 2000). A K Peters, Natick, MA, 2002, pp. 127143.
[BRVD03] Boyd, D. W., Rodriguez-Villegas, F. and Dunfield, N.. Mahler’s measure and the dilogarithm (II). Preprint, 2003, arXiv:0308041v2.
[BV13] Bergeron, N. and Venkatesh, A.. The asymptotic growth of torsion homology for arithmetic groups. J. Inst. Math. Jussieu 12(2) (2013), 391447.
[Den09] Deninger, C.. p-adic entropy and a p-adic Fuglede–Kadison determinant. Algebra, Arithmetic, and Geometry: in Honor of Yu. I. Manin. Vol. I (Progress in Mathematics, 269) . Birkhäuser Boston, Boston, MA, 2009, pp. 423442.
[DGB14] Dikranjan, D. and Giordano Bruno, A.. The bridge theorem for totally disconnected LCA groups. Topology Appl. 169 (2014), 2132.
[EW99] Everest, G. and Ward, T.. Heights of Polynomials and Entropy in Algebraic Dynamics (Universitext) . Springer, London, 1999.
[Fox56] Fox, R. H.. Free differential calculus. III. Subgroups. Ann. of Math. (2) 64 (1956), 407419.
[GAS91] González-Acuña, F. and Short, H.. Cyclic branched coverings of knots and homology spheres. Rev. Mat. Univ. Complut. Madrid 4(1) (1991), 97120.
[GBV15] Giordano Bruno, A. and Virili, S.. Algebraic Yuzvinski formula. J. Algebra 423 (2015), 114147.
[Hil12] Hillman, J.. Algebraic Invariants of Links (Series on Knots and Everything, 52) , 2nd edn. World Scientific, Hackensack, NJ, 2012.
[HMM06] Hillman, J., Matei, D. and Morishita, M.. Pro-p link groups and p-homology groups. Primes and Knots (Contemporary Mathematics, 416) . American Mathematical Society, Providence, RI, 2006, pp. 121136.
[Iwa59] Iwasawa, K.. On 𝛤-extensions of algebraic number fields. Bull. Amer. Math. Soc. (N.S.) 65 (1959), 183226.
[KM08] Kadokami, T. and Mizusawa, Y.. Iwasawa type formula for covers of a link in a rational homology sphere. J. Knot Theory Ramifications 17(10) (2008), 11991221.
[KM13] Kadokami, T. and Mizusawa, Y.. On the Iwasawa invariants of a link in the 3-sphere. Kyushu J. Math. 67(1) (2013), 215226.
[Lal03] Lalín, M. N.. Some examples of Mahler measures as multiple polylogarithms. J. Number Theory 103(1) (2003), 85108.
[Lal04] Lalín, M. N.. Mahler measure and volumes in hyperbolic space. Geom. Dedicata 107 (2004), 211234.
[Le14] Le, T.. Homology torsion growth and Mahler measure. Comment. Math. Helv. 89(3) (2014), 719757.
[LW88] Lind, D. A. and Ward, T.. Automorphisms of solenoids and p-adic entropy. Ergod. Th. & Dynam. Sys. 8(3) (1988), 411419.
[Mih12] Mihara, T.. Singular homology of non-archimedean analytic spaces and integration along cycles. Preprint, 2012, arXiv:1211.1422v1.
[MM82] Mayberry, J. P. and Murasugi, K.. Torsion-groups of abelian coverings of links. Trans. Amer. Math. Soc. 271(1) (1982), 143173.
[Nog07] Noguchi, A.. Zeros of the Alexander polynomial of knot. Osaka J. Math. 44(3) (2007), 567577.
[Por04] Porti, J.. Mayberry–Murasugi’s formula for links in homology 3-spheres. Proc. Amer. Math. Soc. 132(11) (2004), 34233431 (electronic).
[Ril90] Riley, R.. Growth of order of homology of cyclic branched covers of knots. Bull. Lond. Math. Soc. 22(3) (1990), 287297.
[Rol76] Rolfsen, D.. Knots and Links (Mathematics Lecture Series, 7) . Publish or Perish, Berkeley, CA, 1976.
[Sak81] Sakuma, M.. On the polynomials of periodic links. Math. Ann. 257(4) (1981), 487494.
[Shn38] Shnirel’man(Schnirelmann), L. G.. Sur les fonctions dans les corps normés et algébriquement fermés. Izv. Akad. Nauk SSSR Ser. Mat. 2(5–6) (1938), 487498 (Russian).
[Smy81] Smyth, C. J.. On measures of polynomials in several variables. Bull. Aust. Math. Soc. 23(1) (1981), 4963.
[SW02] Silver, D. S. and Williams, S. G.. Mahler measure, links and homology growth. Topology 41(5) (2002), 979991.
[Tan17] Tange, R.. Fox formulas for twisted Alexander invariants associated to representations of knot groups over rings of -integers. Preprint, 2017, arXiv.
[Uek16] Ueki, J.. On the Iwasawa 𝜇-invariants of branched Z p -covers. Proc. Japan Acad. Ser. A Math. Sci. 92(6) (2016), 6772.
[Uek17] Ueki, J.. On the Iwasawa invariants for links and Kida’s formula. Internat. J. Math. 28(6) (2017), 1750035, 30.
[Uek18] Ueki, J.. The profinite completions of knot groups determine the Alexander polynomials. Algebr. Geom. Topol., to appear. Preprint, 2018, arXiv:1702.03836, 13 pages.
[Wal82] Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79) . Springer, New York, 1982.
[Was97] Washington, L. C.. Introduction to Cyclotomic Fields (Graduate Texts in Mathematics, 83) , 2nd edn. Springer, New York, 1997.
[Web79] Weber, C.. Sur une formule de R. H. Fox concernant l’homologie des revêtements cycliques. Enseign. Math. (2) 25(3–4) (1980), 261272.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
Please enter your name
Please enter a valid email address
Who would you like to send this to? *