Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-23T07:27:32.185Z Has data issue: false hasContentIssue false
  • English
  • Français

Higher even dimensional Reidemeister torsion for torus knot exteriors

Published online by Cambridge University Press:  25 April 2013

YOSHIKAZU YAMAGUCHI
YOSHIKAZU YAMAGUCHI*
Affiliation:
Department of Mathematics, Akita University1-1 Tegata-Gakuenmachi, Akita, 010-8502, Japan. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the asymptotics of the higher dimensional Reidemeister torsion for torus knot exteriors, which is related to the results by W. Müller and P. Menal–Ferrer and J. Porti on the asymptotics of the Reidemeister torsion and the hyperbolic volumes for hyperbolic 3-manifolds. We show that the sequence of 1/(2N)2) log | Tor(EK; ρ2N)| converges to zero when N goes to infinity where TorEK; ρ2N is the higher dimensional Reidemeister torsion of a torus knot exterior and an acyclic SL2N(ℂ)-representation of the torus knot group. We also give a classification for SL2(ℂ)-representations of torus knot groups, which induce acyclic SL2N(ℂ)-representations.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 155 , Issue 2 , September 2013 , pp. 297 - 305
Copyright
Copyright © Cambridge Philosophical Society 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Johnson, D. A geometric form of Casson's invariant and its connection to Reidemeister torsion. unpublished lecture notes.Google Scholar
[2]Kitano, T. and Morifuji, T.Twisted Alexander polynomials for irreducible SL(2;ℂ)-representations of torus knots. Ann. Sc. Norm. Super. Pisa Cl. Sci. 11 (2012), 395–406.Google Scholar
[3]Menal–Ferrer, P. and Porti, J.. Higher dimensional Reidemeister torsion invariants for cusped hyperbolic 3-manifolds. arXiv:1110.3718.Google Scholar
[4]Menal–Ferrer, P. and Porti, J.. Twisted cohomology for hyperbolic three manifolds. Osaka J. Math. 49 (2012), 741–769.Google Scholar
[5]Müller, W.The asymptotics of the Ray–Singer analytic torsion of hyperbolic 3-manifolds. In Metric and Differential Geometry, The Jeff Cheeger Anniversary Volume. Progress in Math. vol. 297 (Birkhäuser, 2012), pp. 317–352.CrossRefGoogle Scholar
[6]Raghunathan, M. S.On the first cohomology of discrete subgroups of semisimple Lie groups. Amer. J. Math. 87 (1965), 103–139.CrossRefGoogle Scholar
[7]Turaev, V.Introduction to combinatorial torsions. Lectures in Mathematics (Birkhäuser, 2001).CrossRefGoogle Scholar
[8]Yamaguchi, Y. A surgery formula for the asymptotics of the higher dimensional Reidemeister torsion and Seifert fibered spaces. arXiv:1210.8049.Google Scholar