Posted by: matheuscmss | February 3, 2012

SPCS 4

As usual, we’ll start with a quick revision of SPCS 3. Let M be an origami and denote by \Gamma=Aut(M) the (finite) group of its automorphisms. Given a\in \textrm{Irr}_{\mathbb{R}}(\Gamma) a \Gamma-irreducible \mathbb{R}-representation, let V_a be the \Gamma-module associated to a. In this way, we wrote the interesting part H_1^{(0)}(M,\mathbb{R}) of the (first absolute) homology of M as

H_1^{(0)}(M,\mathbb{R})=\bigoplus\limits_{a\in\textrm{Irr}_{\mathbb{R}}(\Gamma)} W_a

where W_a\simeq V_a^{\ell_a} is the isotypical component associated to V_a. Then, we saw that a could have three types (real, complex or quaternionic), and we studied the nature of the group Sp(W_a) of automorphisms of W_a (as \mathbb{R}(\Gamma)-module) preserving the restriction to W_a of the natural (symplectic) intersection form on H_1(M,\mathbb{R}). This is illustrated in the following table:

{a} {\textit{real}} {\textit{complex}} {\textit{quaternionic}}
{\textrm{dim}_{\mathbb{R}}(V_a)} {\textrm{dim}_{\mathbb{R}}(V_a)} {2\textrm{dim}_{\mathbb{C}}(V_a)} {4\textrm{dim}_{\mathbb{H}}(V_a)}
{\ell_a=\frac{\textrm{dim}_{\mathbb{R}}(W_a)}{\textrm{dim}_{\mathbb{R}}(W_a)}} {\ell_a} even {\ell_a\neq 1} {\ell_a=\frac{1}{2}\ell_{\alpha}}
{Sp(W_a)} {Sp(\ell_a,\mathbb{R})} {U_{\mathbb{C}}(p,q)} {U_{\mathbb{H}}(p,q)}

Here, Sp(\ell_a,\mathbb{R}) is the usual symplectic group over \mathbb{R}, while U_{\mathbb{C}}(p,q) and U_{\mathbb{H}}(p,q) resp. are the unitary group of matrices (with coefficients on \mathbb{C} and \mathbb{H} resp.) preserving the Hermitian form

\sum\limits_{m=1}^p \overline{z}_m z_m - \sum\limits_{m=p+1}^{p+q} \overline{z}_m z_m

where p+q=\ell_a. Also, in the case of a quaternionic, \ell_{\alpha} is the multiplicity of the unique complex representation \alpha with a=\{\alpha\}.

After this quick revision, we are ready to discuss (below the fold) the content of J.-C. Yoccoz’s 4th lecture, that is, how to use the information above on the structure of the group Sp(W_a) can be used to discuss Lyapunov exponents of the Kontsevich-Zorich cocycle on the SL(2,\mathbb{R})-orbit of origamis.

-Moduli spaces of translation surfaces-

Let M be a topological surface, and let \Sigma\subset M a finite subset of points, say \Sigma=\{A_1,\dots, A_s\}, s\geq 1. Consider \kappa=(k_1,\dots,k_s) a list of integers k_i\geq 1 such that

\sum\limits_{i=1}^s (k_i-1)=2g-2

Here, we will think that k_i is “attached” to A_i, and sometimes we denote k_i=:k_{A_i}.

Let \textrm{Diff}(M,\Sigma,\kappa) be the group of homeomorphisms f of M preserving \Sigma and \kappa, that is, k_{f(A_i)}=k_{A_i} for each 1\leq i\leq s, and let \textrm{Diff}_0(M,\Sigma,\kappa) be the connected component of the identity element in \textrm{Diff}(M,\Sigma,\kappa). The quotient group \textrm{Mod}(M,\Sigma,\kappa):=\textrm{Diff}(M,\Sigma,\kappa)/\textrm{Diff}_0(M,\Sigma,\kappa) is the so-called modular group.

We denote by \mathcal{T}(M,\Sigma,\kappa) the set of translation structures on (M,\Sigma,\kappa), and we introduce

\mathcal{Q}(M,\Sigma,\kappa)=\mathcal{T}(M,\Sigma,\kappa)/\textrm{Diff}_0(M,\Sigma,\kappa)

the Teichmuller space of translation structures on (M,\Sigma,\kappa), and

\mathcal{M}(M,\Sigma,\kappa)=\mathcal{T}(M,\Sigma,\kappa)/\textrm{Diff}(M,\Sigma,\kappa) = \mathcal{Q}(M,\Sigma,\kappa)/\textrm{Mod}(M,\Sigma,\kappa)

the moduli space of translation structures on (M,\Sigma,\kappa).

-Action of GL(2,\mathbb{R}) on \mathcal{T}, \mathcal{Q} and \mathcal{M}

Given g\in GL(2,\mathbb{R}), and \zeta\in\mathcal{T}(M,\Sigma,\kappa) a translation structure associated to a translation atlas (\varphi_{\alpha}) on M-\Sigma, we define g_*\zeta the translation structure associated to the translation atlas g\circ\varphi_{\alpha}. This gives a GL(2,\mathbb{R})-action on \mathcal{T}(M,\Sigma,\kappa) commuting with the action of \textrm{Diff}(M,\Sigma,\kappa) (as GL(2,\mathbb{R}) acts by post-composition with translation charts while \textrm{Diff}(M,\Sigma,\kappa) acts by pre-composition with translation charts). Hence, it induces GL(2,\mathbb{R})-actions on \mathcal{Q}(M,\Sigma,\kappa) and \mathcal{M}(M,\Sigma,\kappa).

In this language, the Teichmuller geodesic flow T^t is given by the action of the diagonal matrices \left(\begin{array}{cc} e^t & 0 \\ 0 & e^{-t} \end{array}\right).

-Veech group-

Given [\zeta]\in\mathcal{M}(M,\Sigma,\kappa), its Veech group GL(\zeta) is the stabilizer of [\zeta] under the action of GL(2,\mathbb{R}). Observe that g\in GL(\zeta) implies \det g=\pm1 (as our translation surfaces have finite area). This makes it natural to introduce SL(\zeta):=GL(\zeta)\cap SL(2,\mathbb{R}).

-Affine group-

Given a translation structure \zeta, we denote by

\textrm{Aff}(\zeta):=\{f\in \textrm{Diff}(M,\Sigma,\kappa): f \textrm{ is locally affine when read}

\textrm{ in the charts of } \zeta\}

the affine group of \zeta.

Note that we have a well-defined map \textrm{Aff}(\zeta)\ni f\mapsto Df\in GL(2,\mathbb{R}) obtained by looking at the linear part Df of f in the charts of \zeta. This map is part of an exact sequence

1\to \textrm{Aut}(\zeta)\to\textrm{Aff}(\zeta)\stackrel{D}{\to} GL(\zeta)\to 1

Observe also that A\in\textrm{Aff}(\zeta) induces an isomorphism (M,g_*\zeta)\stackrel{\simeq}{\to} (M,\zeta) where g=DA.

Finally, it is possible to check that the natural map Aff(\zeta)\to \textrm{Mod}(M,\Sigma,\kappa) is injective and its image is exactly the set of \varphi\in\textrm{Mod}(M,\Sigma,\kappa) such that \varphi_*([\zeta])\in GL(2,\mathbb{R})\cdot[\zeta].

-Veech surfaces-

Definition. (M,\zeta) is a Veech surface whenever SL(\zeta) is a lattice in SL(2,\mathbb{R}).

Example. Let (M,\zeta) be a reduced origami (i.e., whose periods generate \mathbb{Z}^2). Then, GL(\zeta) is a finite index subgroup of GL(2,\mathbb{Z}), and hence any origami is a Veech surface.

By a result of J. Smillie, SL(2,\mathbb{R})\cdot[\zeta]\subset\mathcal{M}(M,\Sigma,\kappa) is closed if and only if (M,\zeta) is a Veech surface.

Observe that, in the case of Veech surfaces, SL(2,\mathbb{R})\cdot[\zeta] is naturally identified with the unit tangent bundle of the hyperbolic surface \mathcal{H}/SL(\zeta). Here, \mathcal{H} denotes Poincare’s upper-half plane (we avoided the more standard notation \mathbb{H} as we’re currently using this symbol for Hamilton’s quaternions). In the literature, \mathcal{H}/SL(\zeta) is called Teichmuller surface (or Teichmuller curve).

In this language, the restriction of the Teichmuller flow T^t to SL(2,\mathbb{R})\cdot[\zeta] is the geodesic flow on \mathcal{H}/SL(\zeta).

-Kontsevich-Zorich cocycle-

We begin with the following trivial bundle over \mathcal{Q}(M,\Sigma,\kappa) (whose fibers are 2g-dimensional):

\mathcal{Q}(M,\Sigma,\kappa)\times H_1(M,\mathbb{R})

Observe that the modular group \textrm{Mod}(M,\Sigma,\kappa) acts on both factors of the total space of this trivial bundle: the action on \mathcal{Q}(M,\Sigma,\kappa) was already discussed, while the action on the fibers H_1(M,\mathbb{R}) is by pull-back (as usual). In particular, by using this action we can consider the quotient bundle

(\mathcal{Q}(M,\Sigma,\kappa)\times H_1(M,\mathbb{R}))/\textrm{Mod}(M,\Sigma,\kappa)

over \mathcal{M}(M,\Sigma,\kappa). This quotient bundle is a non-trivial bundle known in the literature as Hodge bundle. The Hodge bundle is the phase space of a cocycle (in the sense of Dynamical Systems)

G_{KZ}^t([\zeta],v)=(T^t[\zeta],v)

called Kontsevich-Zorich cocycle. At first sight this cocycle seems to be trivial, but this is far from being true because the Hodge bundle itself is non-trivial!

Observe that G_{KZ}^t is a symplectic cocycle (as the natural intersection form on H_1(M,\mathbb{R}) is preserved).

Furthermore, this cocycle admits plenty of invariant measures. Below we review the construction of some of them.

a. Masur-Veech measures

The so-called period map \mathcal{Q}(M,\Sigma,\kappa)\stackrel{\Theta}{\to}\textrm{Hom}(H_1(M,\Sigma,\mathbb{Z}), \mathbb{C}) given by

\Theta(\zeta)(\gamma)=\int_\gamma\omega \quad \textrm{ for }\gamma\in H_1(M,\Sigma,\mathbb{Z})

(where \omega is the natural holomorphic 1-form associated to \zeta by pull-back dz from the plane through the translation charts) is a local homeomorphism. Since \textrm{Hom}(H_1(M,\Sigma,\mathbb{Z}), \mathbb{C}) is a complex vector space of dimension 2g+s-1 containing a canonical lattice \textrm{Hom}(H_1(M,\Sigma,\mathbb{Z}), \mathbb{Z}\oplus i\mathbb{Z}) (the image under \Theta of origamis), we can induce a Lebesgue measure on \mathcal{M}(M,\Sigma,\kappa) by using \Theta to pull-back the Lebesgue measure on \textrm{Hom}(H_1(M,\Sigma,\mathbb{Z}), \mathbb{C}) normalized so that the canonical lattice \textrm{Hom}(H_1(M,\Sigma,\mathbb{Z}), \mathbb{Z}\oplus i\mathbb{Z}) has covolume 1.

This Lebesgue measure on \mathcal{M}(M,\Sigma,\kappa) is SL(2,\mathbb{R})-invariant, but it has infinite mass. On the other hand, the induced Lebesgue measure \mu_{MV} on the moduli space \mathcal{M}_1(M,\Sigma,\kappa) of translation structures leading to unit area surfaces has better properties (as it was shown by H. Masur and W. Veech independently):

Theorem (H. Masur, W. Veech). The measure \mu_{MV} has finite mass. The Teichmuller flow is ergodic (and mixing) on each connected component of \mathcal{M}_1(M,\Sigma,\kappa).

In the literature, the SL(2,\mathbb{R})-invariant measure \mu_{MV} is called Masur-Veech measure. Of course, the Masur-Veech measure is invariant under the action of SL(2,\mathbb{R}), it is Teichmuller flow invariant (in particular). So, this provides a class of Teichmuller flow invariant measures.

b. Haar measures on Teichmuller surfaces

Let \zeta be a Veech surface. The unit tangent bundle SL(2,\mathbb{R})/SL(\zeta) of the Teichmuller surface \mathcal{H}/SL(\zeta) has a natural finite and SL(2,\mathbb{R})-invariant Haar measure. The Teichmuller flow is ergodic (and mixing) with respect to this measure because we are in the setting of geodesic flows on the unit tangent bundle of a negatively curved surface.

-Oseledets theorem-

Once we fix an ergodic Teichmuller flow invariant finite measure \mu, we can start the statistical study of the Kontsevich-Zorich cocycle with the aid of Oseledets theorem.

Roughly speaking, Oseledets theorem ensures (in our setting) the existence of real numbers \theta_1>\dots>\theta_r (Lyapunov exponents) such that for \mu-almost every [\zeta] one has a decomposition

H_1(M,\mathbb{R})=\bigoplus E(\theta_i,[\zeta])=\bigoplus E(\theta_i)

where

E(\theta_i)=E(\theta_i,[\zeta])=\{v\in H_1(M,\mathbb{R}): \lim\limits_{t\to\pm\infty}\frac{\log\|p_2(G_{KZ}^t([\zeta],v))\|}{t}=\theta_i\}

(and p_2 denotes the projection into the fibers, i.e., second factor, of Hodge bundle).

Here \|.\| is any norm on the fibers of the Hodge bundle such that G_{KZ}^t verifies the following integrability condition: \log\|G_{KZ}^1\|, \log\|G_{KZ}^{-1}\|\in L^1(\mu). In general this integrability condition is a crucial issue for the application of Oseledets theorem, but in the context of the Kontsevich-Zorich cocycle it is possible to prove that it is automatically satisfied: for instance, by the works of G. Forni, \log\|G_{KZ}^1\|, \log\|G_{KZ}^{-1}\|\leq 1 for a particular norm known as the Hodge norm, so that the integrability conditions holds (with respect to the Hodge norm) as soon as \mu has finite mass.

By the symplecticity of G_{KZ}^t one can show that \theta_i=-\theta_{1+r-i} and \textrm{dim}(E(\theta_i)) = \textrm{dim}(E(\theta_{1+r-i})).

Remark. It is not hard to check from definitions that \theta_1=1 and \theta_r=-1.

It was shown by W. Veech that \textrm{dim}(E(\theta_1))=1 in general.

On the other hand, for the Lebesgue/Masur-Veech measure \mu_{MV} on a connected component of \mathcal{M}_1(M,\Sigma,\kappa), G. Forni proved that G_{KZ}^t is hyperbolic in the sense that \theta_i\neq 0 for all i, and A. Avila and M. Viana improved this result (and solved the so-called Kontsevich-Zorich conjecture) by proving that \textrm{dim}E(\theta_i)=1 for all i.

More recently, A. Eskin, M. Kontsevich and A. Zorich developed explicit formulas for the sum of the non-negative Lyapunov exponents with respect to any “good ”SL(2,\mathbb{R})-invariant finite measure.

Our discussion so far concerned the Kontsevich-Zorich cocycle and its Lyapunov exponents with respect to general Teichmuller flow/SL(2,\mathbb{R}) invariant ergodic measures. From now on, we will specialize to the case of origamis.

-Lyapunov exponents associated to origamis-

Let (M,\zeta) be an origami. The decomposition

H_1(M,\mathbb{R}) = H_1^{st}(M,\mathbb{R})\oplus H_1^{(0)}(M,\mathbb{R})

is:

  • constant along SL(2,\mathbb{R})\cdot[\zeta];
  • invariant under the action of the affine group \textrm{Aff}(\zeta);
  • preserved by G_{KZ}^t.

The factor H_1^{st}(M,\mathbb{R}) corresponds to the Lyapunov exponents \theta_1=1 and \theta_r=-1, and the action of the group \textrm{Aff}^+(\zeta) of orientation-preserving elements of \textrm{Aff}(\zeta) factors through \textrm{Aff}^+(\zeta)\to SL(\zeta)\subset SL(2,\mathbb{R}) where SL(2,\mathbb{R}) acts in the standard way on \mathbb{R}^2\simeq H_1(\mathbb{T}^2,\mathbb{R})\simeq H_1^{st}(M,\mathbb{R}). This justifies the superscript “st” on H_1^{st}(M,\mathbb{R}) as it is the factor of homology where we act in a standard way.

Next, we recall the notation \Gamma=Aut(\zeta)(\simeq N/H) from previous lectures. Given A\in \textrm{Aff}(\zeta), we define an automorphism \rho_A of \Gamma by

\rho_A(\gamma)=A\circ\gamma\circ A^{-1}

In this way, we obtain a homomorphism \textrm{Aff}(\zeta)\stackrel{\rho}{\to}\textrm{Aut}(\Gamma). Since \Gamma and, a fortiori, \textrm{Aut}(\Gamma) are finite, we have that

\textrm{Aff}_*(\zeta) := \textrm{Ker}(\rho)

is a finite-index subgroup of \textrm{Aff}(\zeta).

By definition, any A\in\textrm{Aff}_*(\zeta) commutes with the action of \Gamma, so that the induced map

A_*: H_1^{(0)}(M,\mathbb{R})\to H_1^{(0)}(M,\mathbb{R})

is a homomorphism of \Gamma-modules preserving the natural intersection form on H_1^{(0)}(M,\mathbb{R}).

Consider the decomposition

(1) \quad H_1^{(0)}(M,\mathbb{R}) = \bigoplus\limits_{a\in \textrm{Irr}_{\mathbb{R}}(\Gamma)} W_a

into isotypical irreducible representations.

It follows that, for A\in\textrm{Aff}_*(\zeta), one has A_*(W_a)=W_a, and thus A_*|_{W_a}\in Sp(W_a). In other words, we have a natural homomorphism

\textrm{Aff}_*(\zeta)\to\prod\limits_a Sp(W_a)

The decomposition (1) above is constant along the SL(2,\mathbb{R}) orbit of \zeta, so that it is invariant by G_{KZ}^t (as \textrm{Aut}(g_*\zeta)=g\textrm{Aut}(\zeta)g^{-1} and any character is invariant under conjugation). This allows us to  fix once and for all a\in\textrm{Irr}_{\mathbb{R}}(\Gamma), and consider G_{KZ}^t|_{W_a} and its Lyapunov exponents denoted by (1>)\theta_1>\dots>\theta_r (>-1) by a slight abuse of notation.

Let [\zeta_*]\in SL(2,\mathbb{R})\cdot[\zeta] be a generic point (in the sense that Oseledets theorem can be applied to it). We can write

W_a=\bigoplus E(\theta_i)=E(\theta_i,[\zeta_*])

Since \textrm{Aff}_*(\zeta) is a finite index subgroup of \textrm{Aff}(\zeta) and a generic point [\zeta_*] is recurrent, we can take a sequence of times t_n\to\pm\infty as n\to\pm\infty such that T^{t_n}([\zeta_*]) is very close to [\zeta_*] in such a way that by “closing up” the segment \{T^s([\zeta_*]): 0\leq s\leq t_n\} with a bounded path between T^{t_n}([\zeta_*]) and [\zeta_*], the corresponding closed path leads to an affine diffeomorphism A_n\in\textrm{Aff}_*(\zeta). See the figure below.

In this way, since \|A_n\|\sim e^{|t_n|} (as the largest exponent of the Kontsevich-Zorich cocycle is 1) we have

E(\theta_i)=\{v\in W_a: \lim\limits_{n\to\pm\infty}\frac{\log\|A_n v\|}{\log\|A_n\|}=\pm\theta_i\}

Proposition. E(\theta_i) is a \Gamma-module.

Proof. Because \Gamma is a finite group (and the definition of Lyapunov exponents doesn’t depend on the norm), we may assume that the norm \|.\| is \Gamma-invariant (by the usual averaging argument). Since A_n\in\textrm{Aff}_*(\zeta), it commutes with the action \Gamma. It follows that, for any \gamma\in\Gamma,

\log\|A_n\gamma v\|=\log\|\gamma A_nv\|=\log\|A_n v\|

and thus \gamma v\in E(\theta_i) whenever v\in E(\theta_i). \square

This proposition implies that E(\theta_i)\simeq V_a^{\ell(\theta_i)}. Furthermore,

  • \sum\limits_{i=1}^r\ell(\theta_i)=\ell_a (as W_a\simeq V_a^{\ell_a}), and
  • \ell(\theta_{1+r-i})=\ell(-\theta_i)=\ell(\theta_i) (by symplecticity).

It is worth to note that to the best of our knowledge, the following question is open:

Question. Is there an example where \ell(\theta_i)>1 for some \theta_i\neq0.

In general, we have that \textrm{dim}_{\mathbb{R}}(E(\theta_i)) = \ell(\theta_i)\cdot\textrm{dim}_{\mathbb{R}}(V_a), and

  • \textrm{dim}_{\mathbb{R}}(V_a) = 2 \textrm{dim}_{\mathbb{C}}(V_a) when a is complex;
  • \textrm{dim}_{\mathbb{R}}(V_a) = 4 \textrm{dim}_{\mathbb{H}}(V_a) when a is quaternionic.

In fact, while this general discussion is the best one can do concerning the case a real, it is possible to say a little more about Lyapunov exponents in the complex and quaternionic cases. This is the content of the last section of today’s post.

-Lyapunov exponents when a is complex or quaternionic-

Recall that Sp(W_a)\simeq U_{\mathbb{C}}(p,q) or U_{\mathbb{H}}(p,q) when a is complex or quaternionic.

Lemma 1. Let v\in E(\theta_i), v'\in E(\theta_j), where \theta_i+\theta_j\neq 0. Then, \langle v,v'\rangle_{W_a}=0, where \langle .,.\rangle_{W_a} is the Hermitian form invariant by Sp(W_a) of signature (\overline{p}, \overline{q}) with \overline{p}:=\textrm{dim}_{\mathbb{C}/\mathbb{H}}V_a\cdot p and \overline{q}:=\textrm{dim}_{\mathbb{C}/\mathbb{H}}V_a\cdot q. The real part of \langle .,.\rangle_{W_a} considered as a real quadratic form has signature (\textrm{dim}_{\mathbb{R}}V_a\cdot p, \textrm{dim}_{\mathbb{R}}V_a\cdot q).

Proof. Without loss of generality we may assume \theta_i+\theta_j<0 (as the other case is symmetric). This implies that

\langle v,v'\rangle_{W_a}=\langle A_n v,A_n v' \rangle_{W_a}\sim \|A_n\|^{\theta_i}\cdot\|A_n\|^{\theta_j}\stackrel{n\to\infty}{\to}0

so that the proof is complete. \square

Lemma 2. Let Q be a non-degenerate quadratic form of signature (\overline{p},\overline{q}) and let E be a vector subspace such that Q|_{E}=0. Then, \textrm{dim}(E)\leq \min\{p,q\}.

Proof. Without loss of generality we may assume \overline{p}\leq\overline{q} (as the other case is symmetric), and by appropriately choosing a basis,

Q(x)=\sum\limits_{m=1}^{\overline{p}} x_m^2 - \sum\limits_{m=\overline{p}+1}^{\overline{p}+\overline{q}} x_m^2

If \textrm{dim}(E)\geq \overline{p}+1, we notice that there exists v\in E-\{0\} such that v_1=\dots=v_{\overline{p}}=0: indeed, the algebraic operation of annihilating a given coordinate reduces the dimension of E by 1 at most, so that we can find a non-trivial vector whose first \overline{p} coordinates vanish inside any subspace of dimension \overline{p}+1 (at least). By the expression of Q above, this would mean that Q(v)<0, a contradiction with the assumption Q|_{E}=0. \square

In the setting of origamis, we take Q=\textrm{Re}\langle.,.\rangle_{W_a} and

E^+:=\bigoplus\limits_{\theta_i>0}E(\theta_i)

By Lemma 1, Q|_{E^+}=0, so that, by Lemma 2, we have the following corollary

Corollary. \textrm{dim}_{\mathbb{R}}(V_a)\cdot\sum\limits_{\theta_i>0}\ell(\theta_i):=\textrm{dim}E^+\leq \min\{\overline{p},\overline{q}\}.

Of course, a similar statement is true for

E^-:=\bigoplus\limits_{\theta_j<0}E(\theta_j)

In particular, we conclude the following estimate concerning zero Lyapunov exponents:

\ell(0)\geq |p-q|

Partly motivated by this discussion, we have the following questions:

Question. Is there an example where the inequality \ell(0)\geq |p-q| is strict?

Question. When a is real, is there an example with some zero Lyapunov exponent in W_a?

Question. In the case of hyperelliptic origamis (i.e., origamis whose underlying Riemann surface is hyperelliptic), is there a difference in our discussions so far if we replace \Gamma=\textrm{Aut}(\zeta) by

\Gamma=\widetilde{\textrm{Aut}}(\zeta)=\{A\in\textrm{Aff}(\zeta): DA=\pm \textrm{id}\}?

Closing today’s post, we mention that the plan for the next lecture is to apply the discussions of SPCS 1, 2, 3 and 4 to concrete examples (especially regular and quasi-regular origamis).

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Responses

  1. […] Carlos Matheus: Special curves in Hilbert modular surfaces and Lyapunov exponents of Prym eigenforms, SPCS 4 […]


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