Abstract
We describe the structure of hyperelliptic Rauzy diagrams and hyperelliptic Rauzy–Veech groups. In particular, this provides a solution of the hyperelliptic cases of a conjecture of Zorich on the Zariski closure of Rauzy–Veech groups.
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Notes
See the Appendix 1 below for a concrete example.
On the other hand, Zorich conjecture implies Avila–Viana theorem on the pinching and twisting properties for Rauzy–Veech groups. In fact, Zariski density implies the pinching property by the work of Benoist [4], while the twisting property is automatic (because it has to do with minors of matrices). Hence, our proofs of Theorem 1.1 give new proofs of Avila–Viana theorem in the particular case of hyperelliptic Rauzy–Veech groups.
A path is simple if it does not pass more than once through any vertex.
If \(\gamma \) were of bottom type, we would glue \(\mathcal T\) to the top \(\alpha _b\)-side of \(P_\pi \).
This is somewhat related to [13, Proposition 7.7].
The braid group \(B_m\) is the fundamental group of the space \(\mathbb {C}^{\langle m\rangle }\) of configurations of finite subsets of \(\mathbb {C}\) of cardinality m based at an arbitrarily fixed configuration \(*\in \mathbb {C}^{\langle m\rangle }\).
See [6, Sect. 9.2], for instance.
An interesting consequence of this fact is the non-connectedness of the hyperelliptic Teichmüller spaces.
That is, the entries of \(\rho (g)\in Sp(V)\) depend polynomially on the entires of \(g\in SL(2,\mathbb {R})\).
Its eigenvalues are all real with distinct moduli.
\(A(F)\cap F'=\{0\}\) for all A.B-invariant isotropic subspaces \(F\subset V\) and all A.B-invariant coisotropic subspaces \(F'\subset V\) with \(\text {dim}(F)+\text {dim}(F')=4\).
Pinching elements whose characteristic polynomials have the largest possible Galois group among reciprocal integral polynomials (namely, hyperoctahedral groups).
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Communicated by Nalini Anantharaman.
Appendix A: A pinching and twisting group with small Zariski closure
Appendix A: A pinching and twisting group with small Zariski closure
Let \(\rho \) be the third symmetric power of the standard representation of \(SL(2,\mathbb {R})\). In concrete terms, \(\rho \) is constructed as follows. Consider the basis \(\mathcal {B} = \{X^3, X^2Y, XY^2, Y^3\}\) of the space V of homogenous polynomials of degree 3 on two variables X and Y. By letting \(g=\left( \begin{array}{ll} a &{} b \\ c &{} d \end{array}\right) \in SL(2,\mathbb {R})\) act on X and Y as \(g(X)=aX+cY\) and \(g(Y)=bX+dY\), we obtain an induced action \(\rho (g)\) on V whose matrix in the basis \(\mathcal {B}\) is
Note that the faithful representation \(\rho \) is the unique irreducible four-dimensional representation of \(SL(2,\mathbb {R})\). Furthermore, the matrices \(\rho (g)\) preserve the symplectic structure on V associated to the matrix
Indeed, a direct calculation shows that if \(g=\left( \begin{array}{ll} a &{} b \\ c &{} d \end{array}\right) \), then
where \({}^t\rho (g)\) stands for the transpose of \(\rho (g)\).
Denote by \(\mathcal {M}\) the group generated by the matrices
On one hand, the group \(\mathcal {M}\) has small Zariski closure.
Proposition A.1
The group \(\mathcal {M}\) is not Zariski dense in Sp(V).
Proof
Note that \(\rho \) is a polynomial Footnote 9 homomorphism from \(SL(2,\mathbb {R})\) to Sp(V). In particular, it follows that the image \(H=\rho (SL(2,\mathbb {R}))\) of \(\rho \) is Zariski closed in Sp(V): see, e.g., Corollary 4.6.5 in Witte-Morrris book [12]. Since \(SL(2,\mathbb {Z})\) is a Zariski dense subgroup of \(SL(2,\mathbb {R})\) generated by \(\left( \begin{array}{ll} 1 &{} 1 \\ 0 &{} 1 \end{array}\right) \) and \(\left( \begin{array}{ll} 1 &{} 0 \\ 1 &{} 1 \end{array}\right) \), we see that the Zariski closure of the group \(\mathcal {M}\) is \(H=\rho (SL(2,\mathbb {R}))\) (\(\simeq SL(2,\mathbb {R})\) because \(\rho \) is faithful). This proves the proposition (since H is a proper linear algebraic subgroup of Sp(V)). \(\square \)
On the other hand, the group \(\mathcal {M}\) is pinching and twisting in the sense of Avila–Viana (see [3] and [9, Sect. 2]):
Proposition A.2
The matrix \(A.B\in \mathcal {M}\) is a pinching elementFootnote 10 and the matrix \(A\in \mathcal {M}\) is twistingFootnote 11 with respect to the pinching element \(A.B\in \mathcal {M}\).
Proof
The first assertion follows from the fact that
are the eigenvalues of \(A.B\in \mathcal {M}\).
The second assertion is established by the following reasoning. The columns of
consist of eigenvectors of A.B. Thus, \(T=M^{-1}\cdot A\cdot M\) is the matrix of A in the corresponding basis of eigenvectors of A.B. By definition, A is twisting with respect to A.B when all entries of T and all of its \(2\times 2\) minors associated to Lagrangian planes are non-zero. As it turns out, this last property holds because a direct computation reveals that T and the matrix \(T^{\wedge 2}\) of \(2\times 2\) minors are given by:
and
\(\square \)
In summary, these propositions say that \(\mathcal {M}\) is the desired group: it is pinching and twisting, but not Zariski dense in Sp(V).
Remark A.3
Observe that the group \(\rho (SL(2,\mathbb {Z}))\) does not contain Galois-pinchingFootnote 12 elements of Sp(V) in the sense of [9] because \(H=\rho (SL(2,\mathbb {R}))\) has rank 1. Alternatively, this fact can be shown as follows. A straightforward computation reveals that the characteristic polynomial of \(\rho (g)\) is
and, consequently, the eigenvalues of \(\rho (g)\) are
and
Therefore, the Galois group of the characteristic polynomial \(\rho (g)\), \(g\in SL(2,\mathbb {Z})\), is not the largest possible among reciprocal polynomials of degree four.
Remark A.4
It seems unlikely to find pinching and twisting monoids of symplectic matrices which are not Zariski dense in the context of the Kontsevich–Zorich cocycle. Indeed, Filip’s classification theorem [7] says that, modulo finite-index and compact factors, a Kontsevich–Zorich monodromy \(\mathcal {M}_V\) has Zariski closure \(\text {Sp}(V)\), \(\text {SU}(p,q)\), \(\text {SO}^*(2n)\), \(\wedge ^k \text {SU}(p,1)\) or some spin groups. Thus, all matrices in \(\mathcal {M}_V\) have two eigenvalues with the same modulus unless the Zariski closure of \(\mathcal {M}_V\) is isomorphic to \(\text {Sp}(V)\) modulo finite-index and compact factors. It follows that if \(\mathcal {M}_V\) is pinching, then \(\mathcal {M}_V\) is Zariski dense in \(\text {Sp}(V)\) modulo finite-index and compact factors.
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Avila, A., Matheus, C. & Yoccoz, JC. Zorich conjecture for hyperelliptic Rauzy–Veech groups. Math. Ann. 370, 785–809 (2018). https://doi.org/10.1007/s00208-017-1568-5
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DOI: https://doi.org/10.1007/s00208-017-1568-5