Modular symbols for Teichmüller curves

Curtis T. McMullen

doi:10.1515/crelle-2021-0019

Article

Modular symbols for Teichmüller curves

Curtis T. McMullen

Published/Copyright: May 12, 2021

Published by

Become an author with De Gruyter Brill

Author Information Explore this Subject

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2021 Issue 777

Abstract

This paper introduces a space of nonabelian modular symbols 𝒮⁢(V) attached to any hyperbolic Riemann surface V, and applies it to obtain new results on polygonal billiards and holomorphic 1-forms. In particular, it shows the scarring behavior of periodic trajectories for billiards in a regular polygon is governed by a countable set of measures homeomorphic to ωω+1.

Funding statement: Research supported in part by the NSF.

A Modular symbols and the Weil–Petersson metric

For some additional perspective on modular symbols, in this Appendix we give a short proof of:

Theorem A.1.

Let L⊂R denote the set of lengths of all Weil–Petersson geodesics in M1,1 that begin and end at the cusp. Then L¯ is well ordered, and we have

L¯=〈L〉≅ωω.

Here 〈L〉 is the additive semigroup generated by L.

Proof.

Let V=ℳ1,1≅ℍ/SL2⁡(ℤ) be the moduli space of hyperbolic Riemann surface of genus one with one cusp, endowed with the Weil–Petersson metric. It is well known that the corresponding metric on 𝒯1,1≅ℍ is negatively curved, convex, and incomplete; and that its completion is given by

ℍ*=ℍ∪ℙ1⁢(ℚ).

Moreover, ℍ*/SL2⁡(ℤ)≅V¯=ℳ1,1∪{p} is a compact metric space, with a single added point p corresponding to ℳ0,3. Near p, ℳ1,1 is well approximated metrically by the surface of revolution in ℝ3 obtained by spinning the curve y2=x3 about the x-axis. (See e.g. [Wol].)

Let ℓ⁢(x,y) denote the length of the unique Weil–Petersson geodesic in ℍ joining a given pair of distinct points x,y∈ℙ1⁢(ℚ). Since ℓ⁢(g⁢x,g⁢y)=ℓ⁢(x,y) for all g∈SL2⁡(ℤ), this length gives a map

ℓ:𝒮1⁢(V)→ℝ.

Extending the definition to all modular symbols by

ℓ⁢(γ1*⋯*γm)=∑ℓ⁢(γi),

we obtain a functor ℓ:𝒮⁢(V)→ℝ; this means simply that ℓ⁢(σ*τ)=ℓ⁢(σ)+ℓ⁢(τ). Note that L=ℓ⁢(S1⁢(V)).

We now make two geometric observations. Suppose γn→σ=δ1*⋯*δm in 𝒮⁢(V). Then

ℓ⁢(σ)≤lim sup⁡ℓ⁢(γn),

since length can only be lost in the geometric limit. On the other hand, we also have

(A.1)ℓ⁢(σ)>ℓ⁢(γn)

for all n sufficiently large. Indeed, for all n≫0, a representative γn′ of the homotopy class of γn on ℳ1,1 can be obtained by cutting off δi at distance ϵ from p, and then connecting δi to δi+1 with a curve that spirals finitely many times around the cusp. Due to the shape of the cusp in the Weil–Petersson metric, these spirals each add length on the order of ϵ3/2≪ϵ. Choosing ϵ sufficiently small, we obtain

ℓ⁢(γn)≤ℓ⁢(γn′)<ℓ⁢(σ).

Combining these two observations, we find ℓ⁢(γn)→ℓ⁢(σ) as n→∞. It follows easily that the functor ℓ:𝒮⁢(V)→ℝ is continuous, and comparison to hyperbolic length shows that ℓ is proper.

Let S=⋃d=1∞𝒮d⁢(V). By basic properties of modular symbols (Section 2), we have

S=𝒮1⁢(V)¯=〈𝒮1⁢(V)〉;

and hence, by the properties of ℓ just established, we also have L¯=〈L〉. Equation (A.1) implies that L¯ is well ordered.

It remains to show that L¯ is homeomorphic to ωω; equivalently, that D∞⁢(L¯)=∅ but Dn⁢(L¯)≠∅ for all finite n. The first point follows from the fact that the functor ℓ is proper and D∞⁢(S)=∅; while the second follows from equation (A.1), which implies that Dn⁢(L¯) contains ℓ⁢(𝒮n+1⁢(V)). ∎

Remark.

A related result, valid for all ℳg,n, is announced in [BB, Theorem 1.5].

References

[Bi] B. J. Birch, Elliptic curves over Q: A progress report, 1969 Number Theory Institute (Stony Brook 1969), Proc. Sympos. Pure Math. 20, American Mathematical Society, Providence (1971), 396–400. 10.1090/pspum/020/0314845Search in Google Scholar

[BM] D. W. Boyd and R. D. Mauldin, The order type of the set of Pisot numbers, Topology Appl. 69 (1996), no. 2, 115–120. 10.1016/0166-8641(95)00029-1Search in Google Scholar

[BB] J. F. Brock and K. W. Bromberg, Inflexibility, Weil–Peterson distance, and volumes of fibered 3-manifolds, Math. Res. Lett. 23 (2016), no. 3, 649–674. 10.4310/MRL.2016.v23.n3.a4Search in Google Scholar

[DL] D. Davis and S. Lelièvre, Periodic paths on the pentagon, double pentagon and golden L, preprint (2018), https://arxiv.org/abs/1810.11310. Search in Google Scholar

[Fo] G. Forni, A geometric criterion for the nonuniform hyperbolicity of the Kontsevich–Zorich cocycle. With an appendix by Carlos Matheus, J. Mod. Dyn. 5 (2011), no. 2, 355–395. 10.3934/jmd.2011.5.355Search in Google Scholar

[FMZ] G. Forni, C. Matheus and A. Zorich, Square-tiled cyclic covers, J. Mod. Dyn. 5 (2011), no. 2, 285–318. 10.3934/jmd.2011.5.285Search in Google Scholar

[HM] F. P. Gardiner and H. Masur, Extremal length geometry of Teichmüller space, Complex Variables Theory Appl. 16 (1991), no. 2–3, 209–237. 10.1080/17476939108814480Search in Google Scholar

[GJ] E. Gutkin and C. Judge, Affine mappings of translation surfaces: Geometry and arithmetic, Duke Math. J. 103 (2000), no. 2, 191–213. 10.1215/S0012-7094-00-10321-3Search in Google Scholar

[HS] F. Herrlich and G. Schmithüsen, An extraordinary origami curve, Math. Nachr. 281 (2008), no. 2, 219–237. 10.1002/mana.200510597Search in Google Scholar

[Ho] W. P. Hooper, Grid graphs and lattice surfaces, Int. Math. Res. Not. IMRN 2013 (2013), no. 12, 2657–2698. 10.1093/imrn/rns124Search in Google Scholar

[HL] P. Hubert and E. Lanneau, Veech groups without parabolic elements, Duke Math. J. 133 (2006), no. 2, 335–346. 10.1215/S0012-7094-06-13326-4Search in Google Scholar

[KS] R. Kenyon and J. Smillie, Billiards on rational-angled triangles, Comment. Math. Helv. 75 (2000), no. 1, 65–108. 10.1007/s000140050113Search in Google Scholar

[La] S. Lang, Introduction to modular forms. With appendixes by D. Zagier and Walter Feit. Corrected reprint of the 1976 original, Grundlehren Math. Wiss. 222, Springer, Berlin 1995. 10.1007/978-3-642-51447-0_1Search in Google Scholar

[Lei] C. J. Leininger, On groups generated by two positive multi-twists: Teichmüller curves and Lehmer’s number, Geom. Topol. 8 (2004), 1301–1359. 10.2140/gt.2004.8.1301Search in Google Scholar

[Le] G. Levitt, Foliations and laminations on hyperbolic surfaces, Topology 22 (1983), no. 2, 119–135. 10.1016/0040-9383(83)90023-XSearch in Google Scholar

[Man] Y. I. Manin, Lectures on modular symbols, Arithmetic geometry, Clay Math. Proc. 8, American Mathematical Society, Providence (2009), 137–152. Search in Google Scholar

[Mas] H. Masur, Ergodic theory of translation surfaces, Handbook of dynamical systems. Vol. 1B, Elsevier, Amsterdam (2006), 527–547. 10.1016/S1874-575X(06)80032-9Search in Google Scholar

[MT] K. Matsuzaki and M. Taniguchi, Hyperbolic manifolds and Kleinian groups, Oxford Math. Monogr., Oxford University Press, New York 1998. 10.1093/oso/9780198500629.001.0001Search in Google Scholar

[Maz] B. Mazur, Courbes elliptiques et symboles modulaires, Séminaire Bourbaki. 24ème année (1971/1972), Lecture Notes in Math. 317, Springer, Berlin (1973), Exp. No. 414, 277–294. 10.1007/BFb0069287Search in Google Scholar

[Mc1] C. T. McMullen, Billiards and Teichmüller curves on Hilbert modular surfaces, J. Amer. Math. Soc. 16 (2003), no. 4, 857–885. 10.1090/S0894-0347-03-00432-6Search in Google Scholar

[Mc2] C. T. McMullen, Prym varieties and Teichmüller curves, Duke Math. J. 133 (2006), no. 3, 569–590. 10.1215/S0012-7094-06-13335-5Search in Google Scholar

[Mc3] C. T. McMullen, Braid groups and Hodge theory, Math. Ann. 355 (2013), no. 3, 893–946. 10.1007/s00208-012-0804-2Search in Google Scholar

[Mc4] C. T. McMullen, Cascades in the dynamics of measured foliations, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), no. 1, 1–39. 10.24033/asens.2237Search in Google Scholar

[Mc5] C. T. McMullen, Teichmüller dynamics and unique ergodicity via currents and Hodge theory, J. reine angew. Math. 768 (2020), 39–54. 10.1515/crelle-2019-0037Search in Google Scholar

[Mc6] C. T. McMullen, Billiards, heights and the arithmetic of non-arithmetic groups, preprint (2020). 10.1007/s00222-022-01101-4Search in Google Scholar

[Mo1] M. Möller, Affine groups of flat surfaces, Handbook of Teichmüller theory. Vol. II, IRMA Lect. Math. Theor. Phys. 13, European Mathematical Society, Zürich (2009), 369–387. 10.4171/055-1/11Search in Google Scholar

[Mo2] M. Möller, Shimura and Teichmüller curves, J. Mod. Dyn. 5 (2011), no. 1, 1–32. 10.3934/jmd.2011.5.1Search in Google Scholar

[MS] S. Mozes and N. Shah, On the space of ergodic invariant measures of unipotent flows, Ergodic Theory Dynam. Systems 15 (1995), no. 1, 149–159. 10.1017/S0143385700008282Search in Google Scholar

[Mun] J. R. Munkres, Elementary algebraic topology, Addison-Wesley, Reading 1984. Search in Google Scholar

[Th1] W. P. Thurston, Geometry and topology of three-manifolds, Lecture notes, Princeton University, Princeton (1979). Search in Google Scholar

[Th2] W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431. 10.1090/S0273-0979-1988-15685-6Search in Google Scholar

[Th3] W. P. Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton Math. Ser. 35, Princeton University Press, Princeton 1997. 10.1515/9781400865321Search in Google Scholar

[Th4] W. P. Thurston, Entropy in dimension one, Frontiers in complex dynamics, Princeton Math. Ser. 51, Princeton University Press, Princeton (2014), 339–384. 10.1515/9781400851317-016Search in Google Scholar

[V1] W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math. 97 (1989), no. 3, 553–583. 10.1007/BF01388890Search in Google Scholar

[V2] W. A. Veech, The billiard in a regular polygon, Geom. Funct. Anal. 2 (1992), no. 3, 341–379. 10.1007/BF01896876Search in Google Scholar

[Wol] S. A. Wolpert, Geometry of the Weil–Petersson completion of Teichmüller space, Surveys in differential geometry, Vol. VIII (Boston 2002), Surv. Differ. Geom. 8, International Press, Somerville (2003), 357–393. 10.4310/SDG.2003.v8.n1.a13Search in Google Scholar

[Z] A. Zorich, Flat surfaces, Frontiers in number theory, physics, and geometry. I, Springer, Berlin (2006), 437–583. 10.1007/978-3-540-31347-2_13Search in Google Scholar

Received: 2020-04-03

Revised: 2021-02-25

Published Online: 2021-05-12

Published in Print: 2021-08-01

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/crelle-2021-0019