Algebraic & Geometric Topology

The profinite completions of knot groups determine the Alexander polynomials

Jun Ueki

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Abstract

We study several properties of the completed group ring ̂ [ [ t ̂ ] ] and the completed Alexander modules of knots. Then we prove that if the profinite completions of the groups of two knots J and K are isomorphic, then their Alexander polynomials Δ J ( t ) and Δ K ( t ) coincide.

Article information

Source
Algebr. Geom. Topol., Volume 18, Number 5 (2018), 3013-3030.

Dates
Received: 24 September 2017
Revised: 21 February 2018
Accepted: 5 March 2018
First available in Project Euclid: 30 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.agt/1535594429

Digital Object Identifier
doi:10.2140/agt.2018.18.3013

Mathematical Reviews number (MathSciNet)
MR3848406

Zentralblatt MATH identifier
06935827

Subjects
Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 20E18: Limits, profinite groups 20E26: Residual properties and generalizations; residually finite groups 57M12: Special coverings, e.g. branched

Keywords
profinite completion profinite group ring knot branched covering

Citation

Ueki, Jun. The profinite completions of knot groups determine the Alexander polynomials. Algebr. Geom. Topol. 18 (2018), no. 5, 3013--3030. doi:10.2140/agt.2018.18.3013. https://projecteuclid.org/euclid.agt/1535594429


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References

  • M Artin, B Mazur, On periodic points, Ann. of Math. 81 (1965) 82–99
  • M Asada, On Galois groups of abelian extensions over maximal cyclotomic fields, Tohoku Math. J. 60 (2008) 135–147
  • M F Atiyah, I G Macdonald, Introduction to commutative algebra, Addison-Wesley, Reading, MA (1969)
  • N Bergeron, A Venkatesh, The asymptotic growth of torsion homology for arithmetic groups, J. Inst. Math. Jussieu 12 (2013) 391–447
  • M Boileau, S Boyer, On the Tits alternative for $\mathrm{PD}(3)$ groups, preprint (2017)
  • M Boileau, S Friedl, The profinite completion of $3$–manifold groups, fiberedness and the Thurston norm, preprint (2015)
  • M Boileau, S Friedl, Grothendieck rigidity of $3$–manifold groups, preprint (2017)
  • O Braunling, Torsion homology growth beyond asymptotics, preprint (2017)
  • M R Bridson, F J Grunewald, Grothendieck's problems concerning profinite completions and representations of groups, Ann. of Math. 160 (2004) 359–373
  • M R Bridson, A W Reid, Profinite rigidity, fibering, and the figure-eight knot, preprint (2015)
  • M R Bridson, A W Reid, H Wilton, Profinite rigidity and surface bundles over the circle, Bull. Lond. Math. Soc. 49 (2017) 831–841
  • R H Fox, Free differential calculus, III: Subgroups, Ann. of Math. 64 (1956) 407–419
  • R H Fox, Covering spaces with singularities, from “A symposium in honor of S Lefschetz” (R H Fox, D C Spencer, A W Tucker, editors), Princeton Univ. Press (1957) 243–257
  • D Fried, Cyclic resultants of reciprocal polynomials, from “Holomorphic dynamics” (X Gómez-Mont, J Seade, A Verjovsky, editors), Lecture Notes in Math. 1345, Springer (1988) 124–128
  • S Friedl, S Vidussi, Twisted Alexander polynomials detect fibered $3$–manifolds, Ann. of Math. 173 (2011) 1587–1643
  • L Funar, Torus bundles not distinguished by TQFT invariants, Geom. Topol. 17 (2013) 2289–2344
  • F Q Gouvêa, $p$–adic numbers: an introduction, 2nd edition, Springer (1997)
  • A Grothendieck, Représentations linéaires et compactification profinie des groupes discrets, Manuscripta Math. 2 (1970) 375–396
  • J Hempel, Residual finiteness for $3$–manifolds, from “Combinatorial group theory and topology” (S M Gersten, J R Stallings, editors), Ann. of Math. Stud. 111, Princeton Univ. Press (1987) 379–396
  • J Hempel, Some $3$–manifold groups with the same finite quotients, preprint (2014)
  • C J Hillar, Cyclic resultants, J. Symbolic Comput. 39 (2005) 653–669
  • A Jaikin-Zapirain, Recognition of being fibred for compact $3$–manifolds, preprint (2017) Available at \setbox0\makeatletter\@url http://verso.mat.uam.es/~andrei.jaikin/preprints/fibering.pdf {\unhbox0
  • U Jannsen, Continuous étale cohomology, Math. Ann. 280 (1988) 207–245
  • T Kitayama, M Morishita, R Tange, Y Terashima, On certain $L$–functions for deformations of knot group representations, Trans. Amer. Math. Soc. 370 (2018) 3171–3195
  • K Kodama, M Sakuma, Symmetry groups of prime knots up to $10$ crossings, from “Knots 90” (A Kawauchi, editor), de Gruyter, Berlin (1992) 323–340
  • T Le, Growth of homology torsion in finite coverings and hyperbolic volume, preprint (2014)
  • D D Long, A W Reid, Grothendieck's problem for $3$–manifold groups, Groups Geom. Dyn. 5 (2011) 479–499
  • W Lück, Survey on $L^2$–torsion, talk slides, Ventotene (2015) Available at \setbox0\makeatletter\@url https://tinyurl.com/LueckVentotene {\unhbox0
  • B Mazur, Remarks on the Alexander polynomial, unpublished note (1963/1964) Available at \setbox0\makeatletter\@url https://tinyurl.com/Mazurnote {\unhbox0
  • B Mazur, Primes, knots and Po, lecture notes for the conference “Geometry, topology and group theory” in honor of the 80th birthday of Valentin Poenaru (2012) Available at \setbox0\makeatletter\@url http://www.math.harvard.edu/~mazur/papers/Po8.pdf {\unhbox0
  • S Mochizuki, Topics in absolute anabelian geometry, III: Global reconstruction algorithms, J. Math. Sci. Univ. Tokyo 22 (2015) 939–1156
  • M Morishita, Knots and primes: an introduction to arithmetic topology, Springer (2012)
  • M Morishita, Y Takakura, Y Terashima, J Ueki, On the universal deformations for ${\rm SL}_2$–representations of knot groups, Tohoku Math. J. 69 (2017) 67–84
  • G Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint (2002)
  • G Perelman, Ricci flow with surgery on three-manifolds, preprint (2003)
  • G Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, preprint (2003)
  • K A Perko, Jr, On the classification of knots, Proc. Amer. Math. Soc. 45 (1974) 262–266
  • V P Platonov, O I Tavgen', On the Grothendieck problem of profinite completions of groups, Dokl. Akad. Nauk SSSR 288 (1986) 1054–1058 In Russian; translated in Sov. Math. Dokl. 33 (1986) 822–825
  • L Ribes, P Zalesskii, Profinite groups, 2nd edition, Ergeb. Math. Grenzgeb. 40, Springer (2010)
  • D Rolfsen, Knots and links, Mathematics Lecture Series 7, Publish or Perish, Houston, TX (1990)
  • R Tange, Fox formulas for twisted Alexander invariants associated to representations of knot groups over rings of $S$–integers, J. Knot Theory Ramifications 27 (2018) 1850033, 15
  • J Ueki, $p$–adic Mahler measure and $\mathbb{Z}$–covers of links, preprint (2017) To appear in Ergodic Theory Dynam. Systems
  • C Weber, Sur une formule de R H Fox concernant l'homologie des revêtements cycliques, Enseign. Math. 25 (1979) 261–272
  • G Wilkes, Profinite rigidity for Seifert fibre spaces, Geom. Dedicata 188 (2017) 141–163
  • G Wilkes, Relative cohomology theory for profinite groups, preprint (2017)
  • G Wilkes, Profinite completions, cohomology and JSJ decompositions of compact $3$–manifolds, preprint (2018)
  • G Wilkes, Profinite rigidity of graph manifolds and JSJ decompositions of $3$–manifolds, J. Algebra 502 (2018) 538–587
  • G Wilkes, Profinite rigidity of graph manifolds, II: Knots and mapping classes, preprint (2018)
  • H Wilton, P Zalesskii, Profinite properties of graph manifolds, Geom. Dedicata 147 (2010) 29–45
  • H Wilton, P Zalesskii, Distinguishing geometries using finite quotients, Geom. Topol. 21 (2017) 345–384
  • H Wilton, P Zalesskii, Profinite detection of $3$–manifold decompositions, preprint (2017)