April  2013, 7(2): 239-254. doi: 10.3934/jmd.2013.7.239

Infinitely many lattice surfaces with special pseudo-Anosov maps

1. 

Vassar College, Poughkeepsie, NY 12604-0257, United States

2. 

Oregon State University, Corvallis, OR 97331, United States

Received  October 2012 Revised  May 2013 Published  September 2013

We give explicit pseudo-Anosov homeomorphisms with vanishing Sah-Arnoux-Fathi invariant. Any translation surface whose Veech group is commensurable to any of a large class of triangle groups is shown to have an affine pseudo-Anosov homeomorphism of this type. We also apply a reduction to finite triangle groups and thereby show the existence of nonparabolic elements in the periodic field of certain translation surfaces.
Citation: Kariane Calta, Thomas A. Schmidt. Infinitely many lattice surfaces with special pseudo-Anosov maps. Journal of Modern Dynamics, 2013, 7 (2) : 239-254. doi: 10.3934/jmd.2013.7.239
References:
[1]

P. Arnoux, Échanges d'intervalles et flots sur les surfaces, in "Ergodic Theory'' (Sem., Les Plans-sur-Bex, 1980), Monograph. Enseign. Math., 29, Univ. Genève, Geneva, (1981), 5-38.

[2]

_____, "Thèse de 3$^e$ Cycle,'' Université de Reims, 1981.

[3]

P. Arnoux, J. Bernat and X. Bressaud, Geometrical models for substitutions, Exp. Math., 20 (2011), 97-127. doi: 10.1080/10586458.2011.544590.

[4]

P. Arnoux and T. A. Schmidt, Veech surfaces with nonperiodic directions in the trace field, J. Mod. Dyn., 3 (2009), 611-629. doi: 10.3934/jmd.2009.3.611.

[5]

P. Arnoux and J.-C. Yoccoz, Construction de difféomorphismes pseudo-Anosov, C. R. Acad. Sci. Paris Sér. I Math., 292 (1981), 75-78.

[6]

W. Borho, Kettenbrüche im Galoisfeld, Abh. Math. Sem. Univ. Hamburg, 39 (1973), 76-82. doi: 10.1007/BF02992820.

[7]

W. Borho and G. Rosenberger, Eine Bemerkung zur Hecke-Gruppe $G(\lambda )$, Abh. Math. Sem. Univ. Hamburg, 39 (1973), 83-87. doi: 10.1007/BF02992821.

[8]

I. Bouw and M. Möller, Teichmüller curves, triangle groups, and Lyapunov exponents, Ann. of Math. (2), 172 (2010), 139-185. doi: 10.4007/annals.2010.172.139.

[9]

R. M. Burton, C. Kraaikamp and T. A. Schmidt, Natural extensions for the Rosen fractions, Trans. Amer. Math. Soc., 352 (1999), 1277-1298. doi: 10.1090/S0002-9947-99-02442-3.

[10]

K. Calta, Veech surfaces and complete periodicity in genus two, J. Amer. Math. Soc., 17 (2004), 871-908. doi: 10.1090/S0894-0347-04-00461-8.

[11]

K. Calta and T. A. Schmidt, Continued fractions for a class of triangle groups, J. Austral. Math. Soc., 93 (2012), 21-42. doi: 10.1017/S1446788712000651.

[12]

K. Calta and J. Smillie, Algebraically periodic translation surfaces, J. Mod. Dyn., 2 (2008), 209-248. doi: 10.3934/jmd.2008.2.209.

[13]

P. L. Clark and J. Voight, Algebraic curves uniformized by congruence subgroups of triangle groups, preprint, (2011). Available at http://www.cems.uvm.edu/~jvoight/research.

[14]

L. E. Dickson, "Linear Groups: With an Exposition of the Galois Field Theory,'' Dover Publications, Inc., New York, 1958.

[15]

M.-L. Lang, C.-H. Lim and S.-P. Tan, Principal congruence subgroups of the Hecke groups, J. Number Th., 85 (2000), 220-230. doi: 10.1006/jnth.2000.2542.

[16]

E. Hanson, A. Merberg, C. Towse and E. Yudovina, Generalized continued fractions and orbits under the action of Hecke triangle groups, Acta Arith., 134 (2008), 337-348. doi: 10.4064/aa134-4-4.

[17]

P. Hubert and E. Lanneau, Veech groups without parabolic elements, Duke Math. J., 133 (2006), 335-346. doi: 10.1215/S0012-7094-06-13326-4.

[18]

P. Hubert and T. A. Schmidt, Invariants of translation surfaces, Ann. Inst. Fourier (Grenoble), 51 (2001), 461-495. doi: 10.5802/aif.1829.

[19]

J. H. Lowenstein, G. Poggiaspalla and F. Vivaldi, Interval-exchange transformations over algebraic number fields: The cubic Arnoux-Yoccoz model, Dyn. Syst., 22 (2007), 73-106. doi: 10.1080/14689360601028126.

[20]

_____, Geometric representation of interval-exchange maps over algebraic number fields, Nonlinearity, 21 (2008), 149-177. doi: 10.1088/0951-7715/21/1/009.

[21]

W. P. Hooper, Grid graphs and lattice surfaces, preprint, arXiv:0811.0799, (2009).

[22]

R. Kenyon and J. Smillie, Billiards in rational-angled triangles, Comment. Mathem. Helv., 75 (2000), 65-108. doi: 10.1007/s000140050113.

[23]

A. Leutbecher, Über die Heckeschen Gruppen G($\lambda$), Abh. Math. Sem. Hamb., 31 (1967), 199-205.

[24]

D. Long and A. Reid, Pseudomodular surfaces, J. Reine Angew. Math., 552 (2002), 77-100. doi: 10.1515/crll.2002.094.

[25]

A. M. Macbeath, Generators of linear fractional groups, in "Number Theory'' (Proc. Symp. in Pure Math., Vol. XII, Houston, Tex., 1967) (eds. W. J. Leveque and E. G. Straus), Amer. Math. Soc., Providence, (1969), 14-32.

[26]

C. Maclachlan and A. Reid, "The Arithmetic of Hyperbolic 3-Manifolds,'' Graduate Texts in Mathematics, 219, Springer-Verlag, New York, 2003.

[27]

C. T. McMullen, Teichmüller geodesics of infinite complexity, Acta Math., 191 (2003), 191-223. doi: 10.1007/BF02392964.

[28]

_____, Cascades in the dynamics of measured foliations, preprint, (2012). Available from: http://www.math.harvard.edu/~ctm/papers/.

[29]

T. A. Schmidt and K. M. Smith, Galois orbits of principal congruence Hecke curves, J. London Math. Soc. (2), 67 (2003), 673-685. doi: 10.1112/S0024610703004113.

[30]

L. A. Parson, Normal congruence subgroups of the Hecke groups $G(2^{1/2})$ and $G(3^{1/2})$, Pacific J. Math., 70 (1977), 481-487. doi: 10.2140/pjm.1977.70.481.

[31]

W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.), 19 (1988), 417-431. doi: 10.1090/S0273-0979-1988-15685-6.

[32]

W. A. Veech, Teichmüller curves in modular space, Eisenstein series, and an application to triangular billiards, Inv. Math., 97 (1989), 553-583. doi: 10.1007/BF01388890.

[33]

C. Ward, Calculation of Fuchsian groups associated to billiards in a rational triangle, Ergodic Theory Dynam. Systems, 18 (1998), 1019-1042. doi: 10.1017/S0143385798117479.

[34]

A. Wright, Schwartz triangle mappings and Teichmüller curves: The Veech-Ward-Bouw-Möller curves, Geom. Funct. Anal., 23 (2013), 776-809. doi: 10.1007/s00039-013-0221-z.

show all references

References:
[1]

P. Arnoux, Échanges d'intervalles et flots sur les surfaces, in "Ergodic Theory'' (Sem., Les Plans-sur-Bex, 1980), Monograph. Enseign. Math., 29, Univ. Genève, Geneva, (1981), 5-38.

[2]

_____, "Thèse de 3$^e$ Cycle,'' Université de Reims, 1981.

[3]

P. Arnoux, J. Bernat and X. Bressaud, Geometrical models for substitutions, Exp. Math., 20 (2011), 97-127. doi: 10.1080/10586458.2011.544590.

[4]

P. Arnoux and T. A. Schmidt, Veech surfaces with nonperiodic directions in the trace field, J. Mod. Dyn., 3 (2009), 611-629. doi: 10.3934/jmd.2009.3.611.

[5]

P. Arnoux and J.-C. Yoccoz, Construction de difféomorphismes pseudo-Anosov, C. R. Acad. Sci. Paris Sér. I Math., 292 (1981), 75-78.

[6]

W. Borho, Kettenbrüche im Galoisfeld, Abh. Math. Sem. Univ. Hamburg, 39 (1973), 76-82. doi: 10.1007/BF02992820.

[7]

W. Borho and G. Rosenberger, Eine Bemerkung zur Hecke-Gruppe $G(\lambda )$, Abh. Math. Sem. Univ. Hamburg, 39 (1973), 83-87. doi: 10.1007/BF02992821.

[8]

I. Bouw and M. Möller, Teichmüller curves, triangle groups, and Lyapunov exponents, Ann. of Math. (2), 172 (2010), 139-185. doi: 10.4007/annals.2010.172.139.

[9]

R. M. Burton, C. Kraaikamp and T. A. Schmidt, Natural extensions for the Rosen fractions, Trans. Amer. Math. Soc., 352 (1999), 1277-1298. doi: 10.1090/S0002-9947-99-02442-3.

[10]

K. Calta, Veech surfaces and complete periodicity in genus two, J. Amer. Math. Soc., 17 (2004), 871-908. doi: 10.1090/S0894-0347-04-00461-8.

[11]

K. Calta and T. A. Schmidt, Continued fractions for a class of triangle groups, J. Austral. Math. Soc., 93 (2012), 21-42. doi: 10.1017/S1446788712000651.

[12]

K. Calta and J. Smillie, Algebraically periodic translation surfaces, J. Mod. Dyn., 2 (2008), 209-248. doi: 10.3934/jmd.2008.2.209.

[13]

P. L. Clark and J. Voight, Algebraic curves uniformized by congruence subgroups of triangle groups, preprint, (2011). Available at http://www.cems.uvm.edu/~jvoight/research.

[14]

L. E. Dickson, "Linear Groups: With an Exposition of the Galois Field Theory,'' Dover Publications, Inc., New York, 1958.

[15]

M.-L. Lang, C.-H. Lim and S.-P. Tan, Principal congruence subgroups of the Hecke groups, J. Number Th., 85 (2000), 220-230. doi: 10.1006/jnth.2000.2542.

[16]

E. Hanson, A. Merberg, C. Towse and E. Yudovina, Generalized continued fractions and orbits under the action of Hecke triangle groups, Acta Arith., 134 (2008), 337-348. doi: 10.4064/aa134-4-4.

[17]

P. Hubert and E. Lanneau, Veech groups without parabolic elements, Duke Math. J., 133 (2006), 335-346. doi: 10.1215/S0012-7094-06-13326-4.

[18]

P. Hubert and T. A. Schmidt, Invariants of translation surfaces, Ann. Inst. Fourier (Grenoble), 51 (2001), 461-495. doi: 10.5802/aif.1829.

[19]

J. H. Lowenstein, G. Poggiaspalla and F. Vivaldi, Interval-exchange transformations over algebraic number fields: The cubic Arnoux-Yoccoz model, Dyn. Syst., 22 (2007), 73-106. doi: 10.1080/14689360601028126.

[20]

_____, Geometric representation of interval-exchange maps over algebraic number fields, Nonlinearity, 21 (2008), 149-177. doi: 10.1088/0951-7715/21/1/009.

[21]

W. P. Hooper, Grid graphs and lattice surfaces, preprint, arXiv:0811.0799, (2009).

[22]

R. Kenyon and J. Smillie, Billiards in rational-angled triangles, Comment. Mathem. Helv., 75 (2000), 65-108. doi: 10.1007/s000140050113.

[23]

A. Leutbecher, Über die Heckeschen Gruppen G($\lambda$), Abh. Math. Sem. Hamb., 31 (1967), 199-205.

[24]

D. Long and A. Reid, Pseudomodular surfaces, J. Reine Angew. Math., 552 (2002), 77-100. doi: 10.1515/crll.2002.094.

[25]

A. M. Macbeath, Generators of linear fractional groups, in "Number Theory'' (Proc. Symp. in Pure Math., Vol. XII, Houston, Tex., 1967) (eds. W. J. Leveque and E. G. Straus), Amer. Math. Soc., Providence, (1969), 14-32.

[26]

C. Maclachlan and A. Reid, "The Arithmetic of Hyperbolic 3-Manifolds,'' Graduate Texts in Mathematics, 219, Springer-Verlag, New York, 2003.

[27]

C. T. McMullen, Teichmüller geodesics of infinite complexity, Acta Math., 191 (2003), 191-223. doi: 10.1007/BF02392964.

[28]

_____, Cascades in the dynamics of measured foliations, preprint, (2012). Available from: http://www.math.harvard.edu/~ctm/papers/.

[29]

T. A. Schmidt and K. M. Smith, Galois orbits of principal congruence Hecke curves, J. London Math. Soc. (2), 67 (2003), 673-685. doi: 10.1112/S0024610703004113.

[30]

L. A. Parson, Normal congruence subgroups of the Hecke groups $G(2^{1/2})$ and $G(3^{1/2})$, Pacific J. Math., 70 (1977), 481-487. doi: 10.2140/pjm.1977.70.481.

[31]

W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.), 19 (1988), 417-431. doi: 10.1090/S0273-0979-1988-15685-6.

[32]

W. A. Veech, Teichmüller curves in modular space, Eisenstein series, and an application to triangular billiards, Inv. Math., 97 (1989), 553-583. doi: 10.1007/BF01388890.

[33]

C. Ward, Calculation of Fuchsian groups associated to billiards in a rational triangle, Ergodic Theory Dynam. Systems, 18 (1998), 1019-1042. doi: 10.1017/S0143385798117479.

[34]

A. Wright, Schwartz triangle mappings and Teichmüller curves: The Veech-Ward-Bouw-Möller curves, Geom. Funct. Anal., 23 (2013), 776-809. doi: 10.1007/s00039-013-0221-z.

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