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Finite volume properly convex deformations of the figure-eight knot

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Abstract

In this paper we show that some open set of the representations of the fundamental group of figure-eight knot complement found in Ballas (2014) are the holonomies of a family of finite volume properly convex projective structures on the figure-eight knot complement.

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References

  1. Ballas, S.: Deformations of non-compact, projective manifolds. Algebr. Geom. Topol. 14(5), 2595–2625 (2014)

  2. Ballas, S.: Figure-8 deformations. http://www.math.ucsb.edu/~sballas/research/documents/Fig8Deformations.nb (2013)

  3. Ballas, S.: Flexibility and rigidity of three-dimensional convex projective structures. Ph.D. thesis (2013)

  4. Ballas, S., Long, D.D: Constructing thin subgroups commensurable with the figure-eight knot group, to appear in Alg. & Geom. Top. arXiv:1410.8157v1

  5. Baues, Olivier: The deformations of flat affine structures on the two-torus. ArXiv e-prints (December 2011), arXiv:1112.3263

  6. Benoist, Y.: Convexes divisibles III. Ann. Sci. École Norm. Sup. (4) 38(5), 793–832 (2005). MR2195260 (2007b:22011)

    MATH  MathSciNet  Google Scholar 

  7. Burago, D., Burago, Y., Ivanov, S.: A course in Metric Geometry, Graduate Studies in Mathematics, vol. 33. American Mathematical Society, Providence (2001). MR1835418 (2002e:53053)

    Google Scholar 

  8. Busemann, H.: Intrinsic area. Ann. Math. 48(2), 234–267 (1947). MR0020626 (8,573a)

    Article  MathSciNet  Google Scholar 

  9. Canary, R.D., Epstein, D.B.A., Green, P.: Notes on notes of Thurston. In: Analytical and geometric aspects of hyperbolic space, pp. 3–92, Cambridge University Press, Cambridge (1987)

  10. Cooper, D., Long, D.: A generalization of the Epstein-Penner construction to projective manifolds. ArXiv e-prints (July 2013), arXiv:1307.5016

  11. Cooper, D., Long, D.D., Thistlethwaite, M.B.: Computing varieties of representations of hyperbolic 3-manifolds into SL(4,\(\mathbb{R}\)). Exp. Math. 15(3), 291–305 (2006). MR2264468 (2007k:57032)

    Article  MATH  MathSciNet  Google Scholar 

  12. Cooper, D., Long, D.D., Thistlethwaite, M.B.: Flexing closed hyperbolic manifolds. Geom. Topol. 11, 2413–2440 (2007). MR2372851 (2008k:57031)

    Article  MATH  MathSciNet  Google Scholar 

  13. Cooper, D., Long, D.D., Tillmann, S.: On convex projective manifolds and cusps. ArXiv e-prints (September 2011), arXiv:1109.0585

  14. Cooper, D., Long, D.D., Tillmann, S.: Deforming convex projective manifolds (2013) preprint

  15. Danciger, J.: Geometric transitions: from hyperbolic to ads geometry. Ph.D. thesis (2011)

  16. de la Harpe, P. : On Hilbert’s metric for simplices, geometric group theory, Vol. 1 (Sussex, 1991), pp. 97–119 (1993). MR1238518 (94i:52006)

  17. Goldman, W.M.: Geometric structures on manifolds and varieties of representations, geometry of group representations (Boulder. CO 1987). Contemp. Math., vol. 74, Amer. Math. Soc., Providence, RI, pp. 169–198 (1988)

  18. Goldman, W.M.: Convex real projective structures on compact surfaces. J. Differ. Geom. 31(3), 791–845 (1990). MR1053346 (91b:57001)

    MATH  Google Scholar 

  19. Heusener, M., Porti, J.: Infinitesimal projective rigidity under Dehn filling. Geom. Topol. 15(4), 2017–2071 (2011). MR2860986

    Article  MATH  MathSciNet  Google Scholar 

  20. Johnson, D., Millson, J.J.: Deformation spaces associated to compact hyperbolic manifolds. In: Howe, R. (ed). Discrete Groups in Geometry and Analysis (New Haven. Conn. 1984) pp. 48–106 (1987). MR900823 (88j:22010)

  21. Koszul, J.L.: Déformations de connexions localement plates. Ann. Inst. Fourier (Grenoble) 18(1), no. fasc., 103–114 (1968). MR0239529 (39 #886)

  22. Marquis, L.: Around groups in Hilbert Geometry. ArXiv e-prints (March 2013),  arXiv:1303.7099

  23. Marquis, L.: Exemples de variéetés projectives strictement convexes de volume fini en dimension quelconque. Enseign. Math. 58(1–2), 3–47 (2012). MR2985008

    Article  MATH  MathSciNet  Google Scholar 

  24. Ratcliffe, John G: Foundations of Hyperbolic Manifolds, Second, Graduate Texts in Mathematics, vol. 149. Springer, New York (2006). MR2249478 (2007d:57029)

  25. Riley, R.: An elliptical path from parabolic representations to hyperbolic structures, Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977), pp. 99–133 (1979). MR547459 (81e:57011)

  26. Thurston, W.P.:The geometry and topology of three-manifolds, Princeton lecture notes (Unknown Month 1978)

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Acknowledgments

I would like to thank Daryl Cooper for first suggesting that the representations in [1] might correspond to properly convex projective structures and for several helpful conversations throughout this Project. I would also like to acknowledge support from U.S. National Science Foundation Grants DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network). I would also like to thank the referee for several helpful comments and suggestions including a simplified proof of Lemma 2 and for pointing out an incorrect corollary in a previous version.

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  1. Department of Mathematics, University of California Santa Barbara, Santa Barbara, CA, 93106, USA

    Samuel A. Ballas

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  1. Samuel A. Ballas
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Correspondence to Samuel A. Ballas.

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Ballas, S.A. Finite volume properly convex deformations of the figure-eight knot. Geom Dedicata 178, 49–73 (2015). https://doi.org/10.1007/s10711-015-0043-2

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