Algebraic & Geometric Topology

Translation surfaces and the curve graph in genus two

Duc-Manh Nguyen

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Abstract

Let S be a (topological) compact closed surface of genus two. We associate to each translation surface (X,ω) Ω2 = (2) (1,1) a subgraph Ĉcyl of the curve graph of S. The vertices of this subgraph are free homotopy classes of curves which can be represented either by a simple closed geodesic or by a concatenation of two parallel saddle connections (satisfying some additional properties) on X. The subgraph Ĉcyl is by definition GL+(2, )–invariant. Hence it may be seen as the image of the corresponding Teichmüller disk in the curve graph. We will show that Ĉ cyl is always connected and has infinite diameter. The group Aff+(X,ω) of affine automorphisms of (X,ω) preserves naturally Ĉ cyl, we show that Aff+(X,ω) is precisely the stabilizer of Ĉ cyl in Mod(S). We also prove that Ĉ cyl is Gromov-hyperbolic if (X,ω) is completely periodic in the sense of Calta.

It turns out that the quotient of Ĉ cyl by Aff+(X,ω) is closely related to McMullen’s prototypes in the case that (X,ω) is a Veech surface in (2). We finally show that this quotient graph has finitely many vertices if and only if (X,ω) is a Veech surface for (X,ω) in both strata (2) and (1,1).

Article information

Source
Algebr. Geom. Topol., Volume 17, Number 4 (2017), 2177-2237.

Dates
Received: 31 March 2016
Revised: 30 September 2016
Accepted: 27 October 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510841441

Digital Object Identifier
doi:10.2140/agt.2017.17.2177

Mathematical Reviews number (MathSciNet)
MR3685606

Zentralblatt MATH identifier
1373.51006

Subjects
Primary: 51H20: Topological geometries on manifolds [See also 57-XX]
Secondary: 54H15: Transformation groups and semigroups [See also 20M20, 22-XX, 57Sxx]

Keywords
translation surface curve complex Gromov hyperbolicity

Citation

Nguyen, Duc-Manh. Translation surfaces and the curve graph in genus two. Algebr. Geom. Topol. 17 (2017), no. 4, 2177--2237. doi:10.2140/agt.2017.17.2177. https://projecteuclid.org/euclid.agt/1510841441


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References

  • T Aougab, Uniform hyperbolicity of the graphs of curves, Geom. Topol. 17 (2013) 2855–2875
  • M Bestvina, K Fujiwara, Bounded cohomology of subgroups of mapping class groups, Geom. Topol. 6 (2002) 69–89
  • F Bonahon, Bouts des variétés hyperboliques de dimension $3$, Ann. of Math. 124 (1986) 71–158
  • F Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math. 92 (1988) 139–162
  • B,H Bowditch, from “European Congress of Mathematics” (A Laptev, editor), Eur. Math. Soc., Zürich (2005) 103–115
  • B,H Bowditch, Intersection numbers and the hyperbolicity of the curve complex, J. Reine Angew. Math. 598 (2006) 105–129
  • B,H Bowditch, Uniform hyperbolicity of the curve graphs, Pacific J. Math. 269 (2014) 269–280
  • J,F Brock, R,D Canary, Y,N Minsky, The classification of Kleinian surface groups, II: The ending lamination conjecture, Ann. of Math. 176 (2012) 1–149
  • P Buser, Geometry and spectra of compact Riemann surfaces, Progress in Mathematics 106, Birkhäuser, Boston (1992)
  • K Calta, Veech surfaces and complete periodicity in genus two, J. Amer. Math. Soc. 17 (2004) 871–908
  • K Calta, J Smillie, Algebraically periodic translation surfaces, J. Mod. Dyn. 2 (2008) 209–248
  • M Clay, K Rafi, S Schleimer, Uniform hyperbolicity of the curve graph via surgery sequences, Algebr. Geom. Topol. 14 (2014) 3325–3344
  • M Duchin, C,J Leininger, K Rafi, Length spectra and degeneration of flat metrics, Invent. Math. 182 (2010) 231–277
  • B Farb, D Margalit, A primer on mapping class groups, Princeton Mathematical Series 49, Princeton Univ. Press (2012)
  • R,H Gilman, On the definition of word hyperbolic groups, Math. Z. 242 (2002) 529–541
  • U Hamenstädt, Train tracks and the Gromov boundary of the complex of curves, from “Spaces of Kleinian groups” (Y,N Minsky, M Sakuma, C Series, editors), London Math. Soc. Lecture Note Ser. 329, Cambridge Univ. Press (2006) 187–207
  • U Hamenstädt, Geometry of the complex of curves and of Teichmüller space, from “Handbook of Teichmüller theory, I” (A Papadopoulos, editor), IRMA Lect. Math. Theor. Phys. 11, Eur. Math. Soc., Zürich (2007) 447–467
  • U Hamenstädt, Stability of quasi-geodesics in Teichmüller space, Geom. Dedicata 146 (2010) 101–116
  • J,L Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math. 84 (1986) 157–176
  • W,J Harvey, Boundary structure of the modular group, from “Riemann surfaces and related topics: proceedings of the 1978 Stony Brook conference” (I Kra, B Maskit, editors), Ann. of Math. Stud. 97, Princeton Univ. Press (1981) 245–251
  • S Hensel, P Przytycki, R,C,H Webb, $1$–slim triangles and uniform hyperbolicity for arc graphs and curve graphs, J. Eur. Math. Soc. 17 (2015) 755–762
  • P Hubert, E Lanneau, Veech groups without parabolic elements, Duke Math. J. 133 (2006) 335–346
  • P Hubert, T,A Schmidt, Infinitely generated Veech groups, Duke Math. J. 123 (2004) 49–69
  • N,V Ivanov, Automorphism of complexes of curves and of Teichmüller spaces, Internat. Math. Res. Not. 1997 (1997) 651–666
  • R Kenyon, J Smillie, Billiards on rational-angled triangles, Comment. Math. Helv. 75 (2000) 65–108
  • S Kerckhoff, H Masur, J Smillie, Ergodicity of billiard flows and quadratic differentials, Ann. of Math. 124 (1986) 293–311
  • E Klarreich, The boundary at infinity of the curve complex and the relative Teichmüller space, preprint (1999) Available at \setbox0\makeatletter\@url http://www.math.unicaen.fr/~levitt/klarreich.pdf {\unhbox0
  • M Kontsevich, A Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math. 153 (2003) 631–678
  • E Lanneau, D-M Nguyen, Connected components of Prym eigenform loci in genus three, Math. Ann. (online publication April 2017) 41 pages
  • R Lehnert, On the critical exponent of infinitely generated Veech groups, Math. Ann. (online publication August 2016) 42 pages
  • F Luo, Automorphisms of the complex of curves, Topology 39 (2000) 283–298
  • H Masur, Closed trajectories for quadratic differentials with an application to billiards, Duke Math. J. 53 (1986) 307–314
  • H,A Masur, Y,N Minsky, Geometry of the complex of curves, I: Hyperbolicity, Invent. Math. 138 (1999) 103–149
  • H Masur, S Schleimer, The geometry of the disk complex, J. Amer. Math. Soc. 26 (2013) 1–62
  • H Masur, S Tabachnikov, Rational billiards and flat structures, from “Handbook of dynamical systems, 1A” (B Hasselblatt, A Katok, editors), North-Holland, Amsterdam (2002) 1015–1089
  • C,T McMullen, Teichmüller geodesics of infinite complexity, Acta Math. 191 (2003) 191–223
  • C,T McMullen, Teichmüller curves in genus two: discriminant and spin, Math. Ann. 333 (2005) 87–130
  • C,T McMullen, Dynamics of ${\rm SL}_2(\mathbb R)$ over moduli space in genus two, Ann. of Math. 165 (2007) 397–456
  • Y,N Minsky, Teichmüller geodesics and ends of hyperbolic $3$–manifolds, Topology 32 (1993) 625–647
  • M Möller, Affine groups of flat surfaces, from “Handbook of Teichmüller theory, II” (A Papadopoulos, editor), IRMA Lect. Math. Theor. Phys. 13, Eur. Math. Soc., Zürich (2009) 369–387
  • D-M Nguyen, Parallelogram decompositions and generic surfaces in $\mathscr H^{\rm hyp}(4)$, Geom. Topol. 15 (2011) 1707–1747
  • D-M Nguyen, On the topology of $\mathscr{H}(2)$, Groups Geom. Dyn. 8 (2014) 513–551
  • K Rafi, Hyperbolicity in Teichmüller space, Geom. Topol. 18 (2014) 3025–3053
  • M Rees, An alternative approach to the ergodic theory of measured foliations on surfaces, Ergodic Theory Dynamical Systems 1 (1981) 461–488
  • J Smillie, The dynamics of billiard flows in rational polygons, from “Dynamical systems, ergodic theory and applications” (Y,G Sinai, editor), 2nd edition, Encyclopaedia of Mathematical Sciences 100, Springer (2000) 360–382
  • J Smillie, B Weiss, Minimal sets for flows on moduli space, Israel J. Math. 142 (2004) 249–260
  • J Smillie, B Weiss, Characterizations of lattice surfaces, Invent. Math. 180 (2010) 535–557
  • M Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991) 793–821
  • W,A Veech, from “Algorithms, fractals, and dynamics” (Y Takahashi, editor), Plenum (1995) 217–226
  • Y Vorobets, Periodic geodesics on generic translation surfaces, from “Algebraic and topological dynamics” (S Kolyada, Y Manin, T Ward, editors), Contemp. Math. 385, Amer. Math. Soc. (2005) 205–258
  • A Wright, Cylinder deformations in orbit closures of translation surfaces, Geom. Topol. 19 (2015) 413–438
  • A Wright, Translation surfaces and their orbit closures: an introduction for a broad audience, EMS Surv. Math. Sci. 2 (2015) 63–108
  • A Zorich, Flat surfaces, from “Frontiers in number theory, physics, and geometry, I” (P Cartier, B Julia, P Moussa, P Vanhove, editors), Springer (2006) 437–583