The algebraic hull of the Kontsevich–Zorich cocycle

Abstract

We compute the algebraic hull of the Kontsevich–Zorich cocycle over any $ \mathrm {GL}^+_2(\mathbb {R}) $ invariant subvariety of the Hodge bundle, and derive from this finiteness results on such subvarieties.

  • [SGA3] Schémas en groupes (SGA 3). Tome I. Propriétés générales des schémas en groupes, Gille, P. and Polo, P., Eds., Société Mathématique de France, Paris, 2011, vol. 7.
    @book {SGA3, TITLE = {Schémas en groupes ({SGA} 3). {T}ome {I}. {P}ropriétés générales des schémas en groupes},
      SERIES = {Documents Mathématiques (Paris) [Mathematical Documents(Paris)]},
      VOLUME = {7},
      EDITOR = {Gille, Philippe and Polo, Patrick},
      NOTE = {Séminaire de Géométrie Algébrique du Bois Marie 1962--64. [Algebraic Geometry Seminar of Bois Marie 1962--64]; a seminar directed by M. Demazure and A. Grothendieck with the collaboration of M. Artin, J.-E. Bertin, P. Gabriel, M. Raynaud and J-P. Serre; revised and annotated edition of the 1970 French original},
      PUBLISHER = {Société Mathématique de France, Paris},
      YEAR = {2011},
      PAGES = {xxviii+610},
      ISBN = {978-2-85629-323-2},
      MRCLASS = {14L15},
      MRNUMBER = {2867621},
      ZBLNUMBER = {1241.14003},
      }
  • [AEM] Go to document A. Avila, A. Eskin, and M. Möller, "Symplectic and isometric ${ SL}(2,\Bbb R)$-invariant subbundles of the Hodge bundle," J. Reine Angew. Math., vol. 732, pp. 1-20, 2017.
    @ARTICLE{AEM,
      author = {Avila, Artur and Eskin, Alex and Möller, Martin},
      title = {Symplectic and isometric {${\rm SL}(2,\Bbb R)$}-invariant subbundles of the {H}odge bundle},
      journal = {J. Reine Angew. Math.},
      fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
      volume = {732},
      year = {2017},
      pages = {1--20},
      issn = {0075-4102},
      mrclass = {37C85 (32G15)},
      mrnumber = {3717086},
      doi = {10.1515/crelle-2014-0142},
      url = {https://doi.org/10.1515/crelle-2014-0142},
      zblnumber = {06801922},
      }
  • [ANW] Go to document D. Aulicino, D. Nguyen, and A. Wright, "Classification of higher rank orbit closures in $\mathcal H^{{odd}}(4)$," J. Eur. Math. Soc. (JEMS), vol. 18, iss. 8, pp. 1855-1872, 2016.
    @ARTICLE{ANW,
      author = {Aulicino, David and Nguyen, Duc-Manh and Wright, Alex},
      title = {Classification of higher rank orbit closures in {$\mathcal H^{\rm{odd}}(4)$}},
      journal = {J. Eur. Math. Soc. (JEMS)},
      fjournal = {Journal of the European Mathematical Society (JEMS)},
      volume = {18},
      year = {2016},
      number = {8},
      pages = {1855--1872},
      issn = {1435-9855},
      mrclass = {37D40 (32G15)},
      mrnumber = {3518480},
      mrreviewer = {Jayadev S. Athreya},
      doi = {10.4171/JEMS/631},
      url = {https://doi.org/10.4171/JEMS/631},
      zblnumber = {1369.37044},
      }
  • [AW] P. Apisa and A. Wright, Marked points on translation surfaces.
    @MISC{AW,
      author = {Apisa, Paul and Wright, Alex},
      title={Marked points on translation surfaces},
      arxiv = {1708.03411},
      }
  • [BHM] Go to document M. Bainbridge, P. Habegger, and M. Möller, "Teichmüller curves in genus three and just likely intersections in ${\bf G}^n_m\times {\bf G}^n_a$," Publ. Math. Inst. Hautes Études Sci., vol. 124, pp. 1-98, 2016.
    @ARTICLE{BHM,
      author = {Bainbridge, Matt and Habegger, Philipp and Möller, Martin},
      title = {Teichmüller curves in genus three and just likely intersections in {${\bf G}^n_m\times {\bf G}^n_a$}},
      journal = {Publ. Math. Inst. Hautes Études Sci.},
      fjournal = {Publications Mathématiques. Institut de Hautes Études Scientifiques},
      volume = {124},
      year = {2016},
      pages = {1--98},
      issn = {0073-8301},
      mrclass = {14H10 (32G15)},
      mrnumber = {3578914},
      mrreviewer = {Thomas A. Schmidt},
      doi = {10.1007/s10240-016-0084-6},
      url = {https://doi.org/10.1007/s10240-016-0084-6},
      zblnumber = {1357.14038},
      }
  • [BM] Go to document M. Bainbridge and M. Möller, "The Deligne-Mumford compactification of the real multiplication locus and Teichmüller curves in genus 3," Acta Math., vol. 208, iss. 1, pp. 1-92, 2012.
    @ARTICLE{BM,
      author = {Bainbridge, Matt and Möller, Martin},
      title = {The {D}eligne-{M}umford compactification of the real multiplication locus and {T}eichmüller curves in genus 3},
      journal = {Acta Math.},
      fjournal = {Acta Mathematica},
      volume = {208},
      year = {2012},
      number = {1},
      pages = {1--92},
      issn = {0001-5962},
      mrclass = {14G35 (14Dxx 14Hxx)},
      mrnumber = {2910796},
      doi = {10.1007/s11511-012-0074-6},
      url = {https://doi.org/10.1007/s11511-012-0074-6},
      zblnumber = {1250.14014},
      }
  • [Ca] Go to document K. Calta, "Veech surfaces and complete periodicity in genus two," J. Amer. Math. Soc., vol. 17, iss. 4, pp. 871-908, 2004.
    @ARTICLE{Ca,
      author = {Calta, Kariane},
      title = {Veech surfaces and complete periodicity in genus two},
      journal = {J. Amer. Math. Soc.},
      fjournal = {Journal of the American Mathematical Society},
      volume = {17},
      year = {2004},
      number = {4},
      pages = {871--908},
      issn = {0894-0347},
      mrclass = {37D40 (30F60 32G15 37E05 37E15)},
      mrnumber = {2083470},
      mrreviewer = {Christopher M. Judge},
      doi = {10.1090/S0894-0347-04-00461-8},
      url = {https://doi.org/10.1090/S0894-0347-04-00461-8},
      zblnumber = {1073.37032},
      }
  • [EM] Go to document A. Eskin and M. Mirzakhani, "Invariant and stationary measures for the action on moduli space," Publ. Math. Inst. Hautes Études Sci., p. 230, 2018.
    @article{EM,
      author = {Eskin, Alex and Mirzakhani, Maryam},
      TITLE={Invariant and stationary measures for the action on moduli space},
      year={2018},
      month={Apr},
      day = {17},
      journal = {Publ. {M}ath. {I}nst. {H}autes \'{E}tudes {S}ci.},
      fjournal = {Publications mathématiques. Institut de {H}autes \'{E}tudes de lÕIH\Õ{E}S},
      issn = {1618-1913},
      doi = {10.1007/s10240-018-0099-2},
      url = {https://doi.org/10.1007/s10240-018-0099-2},
      note = {published online 17 April 2018},
      pages = {230 pp.},
      }
  • [EMM] Go to document A. Eskin, M. Mirzakhani, and A. Mohammadi, "Isolation, equidistribution, and orbit closures for the ${ SL}(2,\Bbb R)$ action on moduli space," Ann. of Math. (2), vol. 182, iss. 2, pp. 673-721, 2015.
    @ARTICLE{EMM,
      author = {Eskin, Alex and Mirzakhani, Maryam and Mohammadi, Amir},
      title = {Isolation, equidistribution, and orbit closures for the {${\rm SL}(2,\Bbb R)$} action on moduli space},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {182},
      year = {2015},
      number = {2},
      pages = {673--721},
      issn = {0003-486X},
      mrclass = {58D27 (22F10 32G15 37C85 37D40 60B15)},
      mrnumber = {3418528},
      mrreviewer = {Boris Hasselblatt},
      doi = {10.4007/annals.2015.182.2.7},
      url = {https://doi.org/10.4007/annals.2015.182.2.7},
      zblnumber = {1357.37040},
      }
  • [EMMW] A. Eskin, C. McMullen, R. Mukamel, and A. Wright, Billiards, quadrilaterals and moduli spaces.
    @MISC{EMMW,
      author = {Eskin, Alex and McMullen, Curtis and Mukamel, Ronen and Wright, Alex},
      title = {Billiards, quadrilaterals and moduli spaces},
      note = {preprint},
      zblnumber = {},
      }
  • [sfilip_ssimple] Go to document S. Filip, "Semisimplicity and rigidity of the Kontsevich-Zorich cocycle," Invent. Math., vol. 205, iss. 3, pp. 617-670, 2016.
    @ARTICLE{sfilip_ssimple,
      author = {Filip, Simion},
      title = {Semisimplicity and rigidity of the {K}ontsevich-{Z}orich cocycle},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {205},
      year = {2016},
      number = {3},
      pages = {617--670},
      issn = {0020-9910},
      mrclass = {32G20 (14D07 30F60 37D40)},
      mrnumber = {3539923},
      mrreviewer = {Christian Schnell},
      doi = {10.1007/s00222-015-0643-3},
      url = {https://doi.org/10.1007/s00222-015-0643-3},
      zblnumber = {1368.14013},
      }
  • [sfilip_zero] Go to document S. Filip, "Zero Lyapunov exponents and monodromy of the Kontsevich-Zorich cocycle," Duke Math. J., vol. 166, iss. 4, pp. 657-706, 2017.
    @ARTICLE{sfilip_zero,
      author = {Filip, Simion},
      title = {Zero {L}yapunov exponents and monodromy of the {K}ontsevich-{Z}orich cocycle},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {166},
      year = {2017},
      number = {4},
      pages = {657--706},
      issn = {0012-7094},
      mrclass = {37D25 (14D07 32G15 37D40)},
      mrnumber = {3619303},
      mrreviewer = {Jayadev S. Athreya},
      doi = {10.1215/00127094-3715806},
      url = {https://doi.org/10.1215/00127094-3715806},
      zblnumber = {1370.37066},
      }
  • [sfilip_algebraicity] Go to document S. Filip, "Splitting mixed Hodge structures over affine invariant manifolds," Ann. of Math. (2), vol. 183, iss. 2, pp. 681-713, 2016.
    @ARTICLE{sfilip_algebraicity,
      author = {Filip, Simion},
      title = {Splitting mixed {H}odge structures over affine invariant manifolds},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {183},
      year = {2016},
      number = {2},
      pages = {681--713},
      issn = {0003-486X},
      mrclass = {32S35 (14H55 37A20)},
      mrnumber = {3450485},
      mrreviewer = {Jayadev S. Athreya},
      doi = {10.4007/annals.2016.183.2.5},
      url = {https://doi.org/10.4007/annals.2016.183.2.5},
      zblnumber = {1342.14015},
      }
  • [Etale] Go to document E. Freitag and R. Kiehl, Étale Cohomology and the Weil Conjecture, Springer-Verlag, Berlin, 1988, vol. 13.
    @BOOK{Etale,
      author = {Freitag, Eberhard and Kiehl, Reinhardt},
      title = {Étale Cohomology and the {W}eil Conjecture},
      series = {Ergeb. Math. Grenzgeb.},
      volume = {13},
      note = {translated from the German by Betty S. Waterhouse and William C. Waterhouse, with an historical introduction by J. A. Dieudonné},
      publisher = {Springer-Verlag, Berlin},
      year = {1988},
      pages = {xviii+317},
      isbn = {3-540-12175-7},
      mrclass = {14F20 (11G25 14G13)},
      mrnumber = {0926276},
      mrreviewer = {James Milne},
      doi = {10.1007/978-3-662-02541-3},
      url = {https://doi.org/10.1007/978-3-662-02541-3},
      zblnumber = {0643.14012},
      }
  • [Forni_Matheus_survey] Go to document G. Forni and C. Matheus, "Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards," J. Mod. Dyn., vol. 8, iss. 3-4, pp. 271-436, 2014.
    @ARTICLE{Forni_Matheus_survey,
      author = {Forni, Giovanni and Matheus, Carlos},
      title = {Introduction to {T}eichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards},
      journal = {J. Mod. Dyn.},
      fjournal = {Journal of Modern Dynamics},
      volume = {8},
      year = {2014},
      number = {3-4},
      pages = {271--436},
      issn = {1930-5311},
      mrclass = {37D40 (30F10 30F60 32G15 32G20)},
      mrnumber = {3345837},
      mrreviewer = {Jayadev S. Athreya},
      doi = {10.3934/jmd.2014.8.271},
      url = {https://doi.org/10.3934/jmd.2014.8.271},
      zblnumber = {1377.37057},
      }
  • [FMZ_lyap] Go to document G. Forni, C. Matheus, and A. Zorich, "Lyapunov spectrum of invariant subbundles of the Hodge bundle," Ergodic Theory Dynam. Systems, vol. 34, iss. 2, pp. 353-408, 2014.
    @ARTICLE{FMZ_lyap,
      author = {Forni, Giovanni and Matheus, Carlos and Zorich, Anton},
      title = {Lyapunov spectrum of invariant subbundles of the {H}odge bundle},
      journal = {Ergodic Theory Dynam. Systems},
      fjournal = {Ergodic Theory and Dynamical Systems},
      volume = {34},
      year = {2014},
      number = {2},
      pages = {353--408},
      issn = {0143-3857},
      mrclass = {32G15 (14H10 58A14)},
      mrnumber = {3233697},
      mrreviewer = {Fei Yu},
      doi = {10.1017/etds.2012.148},
      url = {https://doi.org/10.1017/etds.2012.148},
      zblnumber = {1290.37002},
      }
  • [FMZ_zero] Go to document G. Forni, C. Matheus, and A. Zorich, "Zero Lyapunov exponents of the Hodge bundle," Comment. Math. Helv., vol. 89, iss. 2, pp. 489-535, 2014.
    @ARTICLE{FMZ_zero,
      author = {Forni, Giovanni and Matheus, Carlos and Zorich, Anton},
      title = {Zero {L}yapunov exponents of the {H}odge bundle},
      journal = {Comment. Math. Helv.},
      fjournal = {Commentarii Mathematici Helvetici. A Journal of the Swiss Mathematical Society},
      volume = {89},
      year = {2014},
      number = {2},
      pages = {489--535},
      issn = {0010-2571},
      mrclass = {37D25 (32G15 37D40)},
      mrnumber = {3225454},
      doi = {10.4171/CMH/325},
      url = {https://doi.org/10.4171/CMH/325},
      zblnumber = {1316.32010},
      }
  • [Forni_deviations] Go to document G. Forni, "Deviation of ergodic averages for area-preserving flows on surfaces of higher genus," Ann. of Math. (2), vol. 155, iss. 1, pp. 1-103, 2002.
    @ARTICLE{Forni_deviations,
      author = {Forni, Giovanni},
      title = {Deviation of ergodic averages for area-preserving flows on surfaces of higher genus},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {155},
      year = {2002},
      number = {1},
      pages = {1--103},
      issn = {0003-486X},
      mrclass = {37A25 (32G15 37C40 37D25 37D40 37E35 53D25)},
      mrnumber = {1888794},
      mrreviewer = {Howard Masur},
      doi = {10.2307/3062150},
      url = {https://doi.org/10.2307/3062150},
      zblnumber = {1034.37003},
      }
  • [Ham] U. Hamenstaedt, Typical and atypical properties of periodic Teichmüller geodesics, 2014.
    @MISC{Ham,
      author = {Hamenstaedt, Ursula},
      title = {Typical and atypical properties of periodic {T}eichmüller geodesics},
      arxiv = {1409.5978},
      year = {2014},
      zblnumber = {},
      }
  • [Hum] J. E. Humphreys, Linear Algebraic Groups, Springer-Verlag, New York, 1975, vol. 21.
    @BOOK{Hum,
      author = {Humphreys, James E.},
      title = {Linear Algebraic Groups},
      note = {Grad. Texts in Math.},
      volume = {21},
      publisher = {Springer-Verlag, New York},
      year = {1975},
      pages = {xiv+247},
      mrclass = {20GXX (14LXX)},
      mrnumber = {0396773},
      mrreviewer = {T. Ono},
      zblnumber = {0325.20039},
      }
  • [LNW] Go to document E. Lanneau, D. Nguyen, and A. Wright, "Finiteness of Teichmüller curves in non-arithmetic rank $1$ orbit closures," Amer. J. Math., vol. 139, iss. 6, pp. 1449-1463, 2017.
    @ARTICLE{LNW,
      author = {Lanneau, Erwan and Nguyen, Duc-Manh and Wright, Alex},
      title = {Finiteness of {T}eichmüller curves in non-arithmetic rank $1$ orbit closures},
      journal = {Amer. J. Math.},
      fjournal = {American Journal of Mathematics},
      volume = {139},
      year = {2017},
      number = {6},
      pages = {1449--1463},
      issn = {0002-9327},
      mrclass = {37D40 (14H40 53A15)},
      mrnumber = {3730926},
      doi = {10.1353/ajm.2017.0036},
      url = {https://doi.org/10.1353/ajm.2017.0036},
      zblnumber = {1382.32011},
      }
  • [Mc] Go to document C. T. McMullen, "Billiards and Teichmüller curves on Hilbert modular surfaces," J. Amer. Math. Soc., vol. 16, iss. 4, pp. 857-885, 2003.
    @ARTICLE{Mc,
      author = {McMullen, Curtis T.},
      title = {Billiards and {T}eichmüller curves on {H}ilbert modular surfaces},
      journal = {J. Amer. Math. Soc.},
      fjournal = {Journal of the American Mathematical Society},
      volume = {16},
      year = {2003},
      number = {4},
      pages = {857--885},
      issn = {0894-0347},
      mrclass = {32G15 (37D50)},
      mrnumber = {1992827},
      mrreviewer = {Richard Kenyon},
      doi = {10.1090/S0894-0347-03-00432-6},
      url = {https://doi.org/10.1090/S0894-0347-03-00432-6},
      zblnumber = {1030.32012},
      }
  • [McM:spin] Go to document C. T. McMullen, "Teichmüller curves in genus two: Discriminant and spin," Math. Ann., vol. 333, iss. 1, pp. 87-130, 2005.
    @ARTICLE{McM:spin,
      author = {McMullen, Curtis T.},
      title = {Teichmüller curves in genus two: {D}iscriminant and spin},
      journal = {Math. Ann.},
      fjournal = {Mathematische Annalen},
      volume = {333},
      year = {2005},
      number = {1},
      pages = {87--130},
      issn = {0025-5831},
      mrclass = {32G15 (30F10 37D50)},
      mrnumber = {2169830},
      mrreviewer = {Thomas A. Schmidt},
      doi = {10.1007/s00208-005-0666-y},
      url = {https://doi.org/10.1007/s00208-005-0666-y},
      zblnumber = {1086.14024},
      }
  • [Mc7] Go to document C. T. McMullen, "Teichmüller curves in genus two: The decagon and beyond," J. Reine Angew. Math., vol. 582, pp. 173-199, 2005.
    @ARTICLE{Mc7,
      author = {McMullen, Curtis T.},
      title = {Teichmüller curves in genus two: {T}he decagon and beyond},
      journal = {J. Reine Angew. Math.},
      fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
      volume = {582},
      year = {2005},
      pages = {173--199},
      issn = {0075-4102},
      mrclass = {32G15},
      mrnumber = {2139715},
      mrreviewer = {Thomas A. Schmidt},
      doi = {10.1515/crll.2005.2005.582.173},
      url = {https://doi.org/10.1515/crll.2005.2005.582.173},
      zblnumber = {1073.32004},
      }
  • [Mc2] C. T. McMullen, "Prym varieties and Teichmüller curves," Duke Math. J., vol. 133, iss. 3, pp. 569-590, 2006.
    @ARTICLE{Mc2,
      author = {McMullen, Curtis T.},
      title = {Prym varieties and {T}eichmüller curves},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {133},
      year = {2006},
      number = {3},
      pages = {569--590},
      issn = {0012-7094},
      mrclass = {32G15 (14H40 37D50 57M50)},
      mrnumber = {2228463},
      mrreviewer = {Thomas A. Schmidt},
      zblnumber = {1099.14018},
      }
  • [McMullen_classification] Go to document C. T. McMullen, "Dynamics of ${ SL}_2(\Bbb R)$ over moduli space in genus two," Ann. of Math. (2), vol. 165, iss. 2, pp. 397-456, 2007.
    @ARTICLE{McMullen_classification,
      author = {McMullen, Curtis T.},
      title = {Dynamics of {${\rm SL}_2(\Bbb R)$} over moduli space in genus two},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {165},
      year = {2007},
      number = {2},
      pages = {397--456},
      issn = {0003-486X},
      mrclass = {32G15 (37C85 37F99)},
      mrnumber = {2299738},
      mrreviewer = {Serge L. Tabachnikov},
      doi = {10.4007/annals.2007.165.397},
      url = {https://doi.org/10.4007/annals.2007.165.397},
      zblnumber = {1131.14027},
      }
  • [MMW] Go to document C. T. McMullen, R. E. Mukamel, and A. Wright, "Cubic curves and totally geodesic subvarieties of moduli space," Ann. of Math. (2), vol. 185, iss. 3, pp. 957-990, 2017.
    @ARTICLE{MMW,
      author = {McMullen, Curtis T. and Mukamel, Ronen E. and Wright, Alex},
      title = {Cubic curves and totally geodesic subvarieties of moduli space},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {185},
      year = {2017},
      number = {3},
      pages = {957--990},
      issn = {0003-486X},
      mrclass = {32G15 (14H10)},
      mrnumber = {3664815},
      mrreviewer = {Christian Frederik Wei\ss },
      doi = {10.4007/annals.2017.185.3.6},
      url = {https://doi.org/10.4007/annals.2017.185.3.6},
      zblnumber = {06731862},
      }
  • [M3] Go to document M. Möller, "Finiteness results for Teichmüller curves," Ann. Inst. Fourier (Grenoble), vol. 58, iss. 1, pp. 63-83, 2008.
    @ARTICLE{M3,
      author = {Möller, Martin},
      title = {Finiteness results for {T}eichmüller curves},
      journal = {Ann. Inst. Fourier (Grenoble)},
      fjournal = {Université de Grenoble. Annales de l'Institut Fourier},
      volume = {58},
      year = {2008},
      number = {1},
      pages = {63--83},
      issn = {0373-0956},
      mrclass = {14H10 (32G15 32G20)},
      mrnumber = {2401216},
      mrreviewer = {Pierre A. Lochak},
      doi = {10.5802/aif.2344},
      zblnumber = {1140.14010},
      }
  • [Masur_Tab] Go to document H. Masur and S. Tabachnikov, "Rational billiards and flat structures," in Handbook of Dynamical Systems, Vol. 1A, North-Holland, Amsterdam, 2002, pp. 1015-1089.
    @INCOLLECTION{Masur_Tab,
      author = {Masur, Howard and Tabachnikov, Serge},
      title = {Rational billiards and flat structures},
      booktitle = {Handbook of Dynamical Systems, {V}ol. 1{A}},
      pages = {1015--1089},
      publisher = {North-Holland, Amsterdam},
      year = {2002},
      mrclass = {37-02 (28D05 30F60 37A25 37D50 37F99)},
      mrnumber = {1928530},
      mrreviewer = {Richard Kenyon},
      doi = {10.1016/S1874-575X(02)80015-7},
      url = {https://doi.org/10.1016/S1874-575X(02)80015-7},
      zblnumber = {1057.37034},
      }
  • [MirWri] Go to document M. Mirzakhani and A. Wright, "The boundary of an affine invariant submanifold," Invent. Math., vol. 209, iss. 3, pp. 927-984, 2017.
    @ARTICLE{MirWri,
      author = {Mirzakhani, Maryam and Wright, Alex},
      title = {The boundary of an affine invariant submanifold},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {209},
      year = {2017},
      number = {3},
      pages = {927--984},
      issn = {0020-9910},
      mrclass = {37D40 (32G15 58A35)},
      mrnumber = {3681397},
      mrreviewer = {Jayadev S. Athreya},
      doi = {10.1007/s00222-017-0722-8},
      url = {https://doi.org/10.1007/s00222-017-0722-8},
      zblnumber = {1378.37069},
      }
  • [MW] Go to document C. Matheus and A. Wright, "Hodge-Teichmüller planes and finiteness results for Teichmüller curves," Duke Math. J., vol. 164, iss. 6, pp. 1041-1077, 2015.
    @ARTICLE{MW,
      author = {Matheus, Carlos and Wright, Alex},
      title = {Hodge-{T}eichmüller planes and finiteness results for {T}eichmüller curves},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {164},
      year = {2015},
      number = {6},
      pages = {1041--1077},
      issn = {0012-7094},
      mrclass = {37D40 (30F60 32G15)},
      mrnumber = {3336840},
      mrreviewer = {Jayadev S. Athreya},
      doi = {10.1215/00127094-2885655},
      url = {https://doi.org/10.1215/00127094-2885655},
      zblnumber = {1345.37029},
      }
  • [Wtot] A. Wright, Totally geodesic submanifolds of Teichmüller space.
    @MISC{Wtot,
      author = {Wright, Alex},
      title = {{T}otally geodesic submanifolds of {T}eichm{ü}ller space},
      note = {forthcoming},
      zblnumber = {},
      }
  • [AW_field] Go to document A. Wright, "The field of definition of affine invariant submanifolds of the moduli space of abelian differentials," Geom. Topol., vol. 18, iss. 3, pp. 1323-1341, 2014.
    @ARTICLE{AW_field,
      author = {Wright, Alex},
      title = {The field of definition of affine invariant submanifolds of the moduli space of abelian differentials},
      journal = {Geom. Topol.},
      fjournal = {Geometry \& Topology},
      volume = {18},
      year = {2014},
      number = {3},
      pages = {1323--1341},
      issn = {1465-3060},
      mrclass = {32G15 (30F60 37D40)},
      mrnumber = {3254934},
      mrreviewer = {Jayadev S. Athreya},
      doi = {10.2140/gt.2014.18.1323},
      url = {https://doi.org/10.2140/gt.2014.18.1323},
      zblnumber = {1320.32019},
      }
  • [Wcyl] Go to document A. Wright, "Cylinder deformations in orbit closures of translation surfaces," Geom. Topol., vol. 19, iss. 1, pp. 413-438, 2015.
    @ARTICLE{Wcyl,
      author = {Wright, Alex},
      title = {Cylinder deformations in orbit closures of translation surfaces},
      journal = {Geom. Topol.},
      fjournal = {Geometry \& Topology},
      volume = {19},
      year = {2015},
      number = {1},
      pages = {413--438},
      issn = {1465-3060},
      mrclass = {32G15 (37D40)},
      mrnumber = {3318755},
      mrreviewer = {Jayadev S. Athreya},
      doi = {10.2140/gt.2015.19.413},
      url = {https://doi.org/10.2140/gt.2015.19.413},
      zblnumber = {1318.32021},
      }
  • [Wsurvey] Go to document A. Wright, "Translation surfaces and their orbit closures: an introduction for a broad audience," EMS Surv. Math. Sci., vol. 2, iss. 1, pp. 63-108, 2015.
    @ARTICLE{Wsurvey,
      author = {Wright, Alex},
      title = {Translation surfaces and their orbit closures: an introduction for a broad audience},
      journal = {EMS Surv. Math. Sci.},
      fjournal = {EMS Surveys in Mathematical Sciences},
      volume = {2},
      year = {2015},
      number = {1},
      pages = {63--108},
      issn = {2308-2151},
      mrclass = {32G15 (30D05 30F30 30F60)},
      mrnumber = {3354955},
      mrreviewer = {Jayadev S. Athreya},
      doi = {10.4171/EMSS/9},
      url = {https://doi.org/10.4171/EMSS/9},
      zblnumber = {1372.37090},
      }
  • [Zimmer_book] Go to document R. J. Zimmer, Ergodic Theory and Semisimple Groups, Birkhäuser Verlag, Basel, 1984, vol. 81.
    @BOOK{Zimmer_book,
      author = {Zimmer, Robert J.},
      title = {Ergodic Theory and Semisimple Groups},
      series = {Monogr. Math.},
      volume = {81},
      publisher = {Birkhäuser Verlag, Basel},
      year = {1984},
      pages = {x+209},
      isbn = {3-7643-3184-4},
      mrclass = {22E40 (22D40 28D15)},
      mrnumber = {0776417},
      mrreviewer = {S. G. Dani},
      doi = {10.1007/978-1-4684-9488-4},
      url = {https://doi.org/10.1007/978-1-4684-9488-4},
      zblnumber = {0571.58015},
      }
  • [Zorich_survey] A. Zorich, "Flat surfaces," in Frontiers in Number Theory, Physics, and Geometry. I, Springer, Berlin, 2006, pp. 437-583.
    @INCOLLECTION{Zorich_survey,
      author = {Zorich, Anton},
      title = {Flat surfaces},
      booktitle = {Frontiers in Number Theory, Physics, and Geometry. {I}},
      pages = {437--583},
      publisher = {Springer, Berlin},
      year = {2006},
      mrclass = {37D40 (30F30 32G15 37D50 57M50)},
      mrnumber = {2261104},
      mrreviewer = {Thomas A. Schmidt},
      zblnumber = {1129.32012},
      }

Authors

Alex Eskin

University of Chicago, Chicago, IL

Simion Filip

Harvard University, Cambridge, MA

Alex Wright

Stanford University, Stanford, CA