Modular symbols for Teichmüller curves

Curtis T. McMullen

doi:10.1515/crelle-2021-0019

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Modular symbols for Teichmüller curves

Curtis T. McMullen

Veröffentlicht/Copyright: 12. Mai 2021

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Aus der Zeitschrift Journal für die reine und angewandte Mathematik (Crelles Journal) Band 2021 Heft 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].

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Received: 2020-04-03

Revised: 2021-02-25

Published Online: 2021-05-12

Published in Print: 2021-08-01

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