Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-28T14:55:53.920Z Has data issue: false hasContentIssue false
  • English
  • Français

Prising apart geodesics by length in hyperbolic manifolds

Published online by Cambridge University Press:  24 March 2015

JAMES W. ANDERSON
JAMES W. ANDERSON*
Affiliation:
Mathematical Sciences, University of Southampton, Southampton, SO17 1BJ. email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We develop a condition on a closed curve on a surface or in a 3-manifold that implies that the length function associated to the curve on the space of all hyperbolic structures on the surface or in the 3-manifold (respectively) completely determines the curve. Specifically, for an orientable surface S of negative Euler characteristic, we extend the known result that simple curves have this property to curves with self-intersection number one (with one exceptional case arising from hyperellipticity that we describe completely). For a large class of hyperbolizable 3-manifolds, we show that curves freely homotopic to simple curves on ∂M have this property.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 158 , Issue 3 , May 2015 , pp. 547 - 572
Copyright
Copyright © Cambridge Philosophical Society 2015 

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]Abikoff, W.The real analytic theory of Teichmüller space. Lecture Notes in Math. 820 (Springer-Verlag, 1976).Google Scholar
[2]Abraham, R.Bumpy metrics. Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, CA, 1968) (Amer. Math. Soc, Providence, RI, 1970) 1–3.Google Scholar
[3]Anderson, J. W.Variations on a theme of Horowitz. In Kleinian Groups and Hyperbolic 3-Manifolds, Lond. Math. Soc. Lec. Notes 299, ed. by Komori, Y., Markovic, V., and Series, C. (Cambridge University Press, 2003), 307–341.Google Scholar
[4]Anderson, J. W., Canary, R. D. and McCullough, D.On the topology of deformation spaces of Kleinian groups. Ann. of Math. 152 (2000), 693–741.Google Scholar
[5]Basmajian, A.The stable neighbourhood theorem and lengths of closed geodesics. Proc. Amer. Math. Soc. 119 (1993), 217–224.Google Scholar
[6]Basmajian, A.Universal length bounds for non-simple closed geodesics on hyperbolic surfaces. J. Topology, 6 (2013), 513–524.Google Scholar
[7]Bright, H. E. Characters and lengths of geodesics in hyperbolic 3-manifolds. PhD. thesis. University of Southampton (2007).Google Scholar
[8]Brock, J. F., Canary, R. D. and Minsky, Y. N.The classification of Kleinian surface groups, II: the ending lamination conjecture. Ann. of Math. (2) 176 (2012), 1–149.CrossRefGoogle Scholar
[9]Buser, P.Geometry and Spectra of Compact Riemann Surfaces. Progr. Math. 106 (Birkhäuser, Basel, 1992).Google Scholar
[10]Canary, R. D. and McCullough, D.Homotopy equivalences of 3-manifolds and deformation theory of Kleinian groups. Mem. Amer. Math. Soc. 172 (American Mathematical Society, 2004).Google Scholar
[11]Comar, T. D. Hyperbolic Dehn surgery and convergence of Kleinian groups. PhD. thesis. University of Michigan (1996).Google Scholar
[12]Farkas, H. M. and Kra, I.Riemann surfaces. Graduate Texts in Mathematics 71 (Springer Verlag, New York, 1992).Google Scholar
[13]Fujii, M.Hyperbolic 3-manifolds with totally geodesic boundary. Osaka J. Math. 27 (1990), 539–553.Google Scholar
[14]Ginzburg, D. and Rudnick, Z.Stable multiplicities in the length spectrum of Riemann surfaces. Israel J. Math. 104 (1998), 129–144.Google Scholar
[15]Haas, A. and Susskind, P.The geometry of the hyperelliptic involution in genus two. Proc. Amer. Math. Soc. 105 (1989), 159–165.Google Scholar
[16]Hempel, J.3-manifolds. Ann. Math. Stud. 86 (Princeton University Press, 1976).Google Scholar
[17]Hempel, J.Traces, lengths, and simplicity of loops on surfaces. Topology Appl. 18 (1984), 153–161.Google Scholar
[18]Horowitz, R. D.Characters of free groups represented in the two-dimensional special linear group. Comm. Pure Appl. Math. 25 (1972), 635–649.Google Scholar
[19]Johannson, K.Homotopy equivalences of 3-manifolds with boundaries. Lecture Notes in Math. vol 761 (Springer, Berlin, 1979).Google Scholar
[20]Jørgensen, T.On discrete groups of Möbius transformations. Amer. J. Math. 98 (1976), 739–749.Google Scholar
[21]Kapovich, M.Hyperbolic manifolds and discrete groups. Progr. Math., vol 183, (Birkhäuser Boston, Inc., Boston, MA, 2001).Google Scholar
[22]Leininger, C. J.Equivalent curves in surfaces. Geom. Dedicata 102 (2003), 151–177.CrossRefGoogle Scholar
[23]Kra, I. On lifting Kleinian groups to SL(2, $\mathbb C$). In Differential Geometry and Complex Analysis, edited by Chavel, I. and Farkas, H. M. (Springer-Verlag, Berlin, 1985), 181–193.Google Scholar
[24]Maskit, B.Intersections of component subgroups of Kleinian groups. In Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973), Ann. Math. Stud. 79 (Princeton University Press, 1974), 349–367.CrossRefGoogle Scholar
[25]Maskit, B.Kleinian groups. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol 287 (Springer-Verlag, Berlin, 1988).Google Scholar
[26]Maskit, B.Parabolic elements in Kleinian groups. Ann. of Math. 117 (1983), 659–668.Google Scholar
[27]Masters, J. D.Length multiplicities in hyperbolic 3-manifolds. Israel J. Math. 119 (2000), 9–28.Google Scholar
[28]Matsuzaki, K. and Taniguchi, M.Hyperbolic Manifolds and Kleinian Groups. Oxford Mathematical Monographs (Oxford Science Publications, Clarendon Press, Oxford University Press, New York, 1998).Google Scholar
[29]McCullough, D.Compact submanifolds of 3-manifolds with boundary. Quart. J. Math. Oxford 37 (1986), 299–307.Google Scholar
[30]McCullough, D., Miller, A. and Swarup, G.Uniqueness of cores of noncompact 3-manifolds. J. London Math. Soc. 32 (1985), 548–556.Google Scholar
[31]McMullen, C. T.Iteration on Teichmüller space. Invent. Math. 99 (1990), 425–454.Google Scholar
[32]McShane, G. Homeomorphisms which preserve simple geodesics, preprint, 1993.Google Scholar
[33]Morgan, J. W.On Thurston's uniformisation theorem for three-dimensional manifolds. In The Smith Conjecture, ed. by Morgan, J. W. and Bass, H. (Academic Press, 1984), 37–125.Google Scholar
[34]Myers, R.Simple knots in compact, orientable 3-manifolds. Trans. Amer. Math. Soc. 273 (1983), 75–92.Google Scholar
[35]Nakanishi, T.The lengths of the closed geodesics on a Riemann surface with self-intersections. Tohoku Math. J. 41 (1989), 527–541.CrossRefGoogle Scholar
[36]Randol, B.The length spectrum of a Riemann surface is always of unbounded multiplicity. Proc. Amer. Math. Soc. 78 (1980), 455–456.Google Scholar
[37]Rivin, I.Geodesics with one self-intersection, and other stories. Adv. Math. 231 (2012), 2391–2412.Google Scholar
[38]Thurston, W. P.Hyperbolic structures on 3-manifolds. I. Deformation of acylindrical manifolds. Ann. of Math. 124 (1986), 203–246.Google Scholar
[39]Yamada, A.On Marden's universal constant of Fuchsian groups. Kodai Math. J. 4 (1981), 266–277.CrossRefGoogle Scholar
[40]Yamada, A.On Marden's universal constant of Fuchsian groups, II. J. d'Analyse Math. 41 (1982), 234–248.Google Scholar