Minimality versus twisting for linear monoids and the geometric meaning of Rauzy-Veech pinching: the open questions from Avila-Viana's proof of the Zorich-Kontsevich conjecture

In 2007 Avila and Viana proved the Zorich-Kontsevich conjecture - simplicity of the Lyapunov spectrum of the Kontsevich-Zorich cocycle on every stratum of Abelian differentials - by reducing it to two algebraic richness properties, pinching and twisting, of the monoid generated by the Rauzy-Veech renormalization matrices; their paper explicitly left three structural questions open. The third question - the Zariski closure of the group generated by the Rauzy monoid (Remark 6.12; the implied Zariski density is Zorich's 1999 conjecture) - is resolved: Avila-Matheus-Yoccoz determined the hyperelliptic Rauzy-Veech groups, and Gutiérrez-Romo classified all of them as explicit finite-index subgroups of $\mathrm{Sp}(2g,\mathbb{Z})$, hence Zariski-dense. The surviving open core consists of the other two questions, stated here source-faithfully: (i) Problem 4.13 of the source - if a monoid of automorphisms of a finite-dimensional real vector space acts minimally on the projective space, is it necessarily twisting; and, for symplectic actions, does minimality on projective space already imply twisting of isotropic subspaces; (ii) Remark 6.11 of the source - find a geometric interpretation, in terms of the Teichmüller flow, of the parabolic behavior that the proof produces by concatenating Rauzy paths.

Background

A translation surface is a compact Riemann surface together with a nonzero holomorphic 1-form (an Abelian differential); away from the zeros of the form the transition maps between natural flat charts are planar translations, so the surface carries a singular flat metric. The moduli space of genus-$g$ translation surfaces decomposes into strata indexed by the orders of the zeros, each carrying a natural Lebesgue-class measure (the Masur-Veech measure) and an action of $\mathrm{SL}(2,\mathbb{R})$ whose diagonal part is the Teichmüller (geodesic) flow. The same objects have a one-dimensional shadow: interval exchange transformations - maps of an interval that cut it into finitely many subintervals and permute them by translations - arise as first-return maps of the vertical foliation of a translation surface. The combinatorics of the Teichmüller flow is captured by Rauzy-Veech induction, an explicit renormalization algorithm on permutations, each step of which acts on the (relative) homology of the surface by an integer Kontsevich-Zorich matrix. Over the renormalization dynamics sits the Kontsevich-Zorich (KZ) cocycle: the linear cocycle describing the induced action on homology, to which the multiplicative ergodic theorem assigns Lyapunov exponents - the exponential growth rates of homology vectors along typical orbits. Zorich discovered that these exponents control the deviation of ergodic averages of typical translation flows and interval exchanges, and with Kontsevich he conjectured that, for the Masur-Veech measure on any connected component of any stratum of Abelian differentials, the nontrivial Lyapunov exponents of the KZ cocycle are all distinct (the spectrum is simple), which organizes the deviation statistics into a complete asymptotic Lagrangian flag. After partial results of Forni, the conjecture was proved in full by Avila and Viana (2007). Their proof replaces the cocycle by a purely algebraic object: the Rauzy monoid, the semigroup of Kontsevich-Zorich matrices generated by the induction, restricted to a suitable symplectic subspace $H(\pi)$ of dimension $2g$ associated with a permutation $\pi$ of the Rauzy class under consideration (symplectic means preserving a non-degenerate skew-symmetric bilinear form $\omega$). In a companion paper they proved a general sufficient criterion: if such a monoid $\mathcal{B}$ is pinching and twisting, the Lyapunov spectrum of the associated cocycle is simple. Twisting is an algebraic transversality condition: writing $\mathrm{Grass}(k,H)$ for the Grassmannian, the set of $k$-dimensional subspaces of $H$, $\mathcal{B}$ twists a subspace $F\in\mathrm{Grass}(k,H)$ if for every finite family $F_1,\dots,F_m$ of subspaces of complementary dimension some element of $\mathcal{B}$ moves $F$ off all of them at once ($x\cdot F\cap F_i=\{0\}$ for all $i$); equivalently, $F$ is trapped in no nontrivial $\mathcal{B}$-invariant linear arrangement (a finite union of projective linear subspaces of the projectivized exterior power, intersected with the Grassmannian). The monoid is twisting if this holds for every proper subspace. Pinching is a spectral-gap condition: the singular values of elements of $\mathcal{B}$ (the square roots of the eigenvalues of $x^{*}x$) can be made arbitrarily spread out. For the actual Rauzy monoids, twisting is proved by first showing that the projective action on $\mathbf{P}H(\pi)$ is minimal - the only nonempty closed subset $C\subset\mathbf{P}H$ with $x\cdot C=C$ for all $x\in\mathcal{B}$ is the whole space - and then upgrading, through an analysis of stabilizers, to minimality on the space of Lagrangian flags (maximal chains of isotropic subspaces, that is, subspaces on which $\omega$ vanishes), from which twisting of isotropic subspaces follows; pinching is obtained by exhibiting explicit elements with parabolic block structure (an eigenvalue 1 with a nontrivial Jordan block in the central part), built by concatenating pairs of Rauzy paths, with genus-changing steps that use the degeneration inverse to 'bubbling a handle' (Lemma 20 of Kontsevich-Zorich). The proof left three structural questions open: whether the purely topological condition of projective minimality already forces twisting (Problem 4.13); a geometric interpretation, in terms of the Teichmüller flow, of the parabolic behavior of the concatenated Rauzy paths (Remark 6.11); and the Zariski closure - the smallest algebraic subgroup, i.e. subgroup defined by polynomial equations, containing the group generated by the monoid (Remark 6.12), which contains Zorich's 1999 conjecture that this closure is the full symplectic group. The logical landscape is now partly mapped: Zariski density implies pinching (a theorem of Benoist) and twisting (a minors argument), but the converse fails - Avila-Matheus-Yoccoz constructed, via the third symmetric power of the standard representation of $\mathrm{SL}(2,\mathbb{Z})$, a group that is pinching and twisting yet has small Zariski closure - and the Zariski-closure question itself has been resolved completely.

Problem Statement

Let $H$ be a finite-dimensional real vector space and $\mathcal{B}\subset\mathrm{GL}(H)$ a submonoid (containing the identity and closed under products). Throughout, $\mathbf{P}H$ denotes the projective space of lines in $H$, on which each $x\in\mathcal{B}$, being a linear isomorphism, acts as a homeomorphism, and $\mathrm{Grass}(k,H)$ denotes the Grassmannian of $k$-dimensional subspaces of $H$. Say that $\mathcal{B}$ acts minimally on the projective space $\mathbf{P}H$ if the only nonempty closed subset $C\subset\mathbf{P}H$ satisfying $x\cdot C=C$ for every $x\in\mathcal{B}$ is $\mathbf{P}H$ itself. Say that $\mathcal{B}$ twists a subspace $F\in\mathrm{Grass}(k,H)$ if for every finite family $F_1,\dots,F_m\in\mathrm{Grass}(\dim H-k,H)$ there exists $x\in\mathcal{B}$ such that $x\cdot F\cap F_i=\{0\}$ for all $i$; $\mathcal{B}$ is twisting if it twists every subspace $F$ of every dimension $1\le k\le\dim H-1$. Equivalently (Avila-Viana's Lemma 4.2), $\mathcal{B}$ is twisting exactly when no subspace $F$ with $1\le k=\dim F\le\dim H-1$ is trapped in a nontrivial $\mathcal{B}$-invariant linear arrangement of $\mathrm{Grass}(k,H)$, that is, a finite union of projective linear subspaces of the projectivized exterior power $\Lambda^k H$ intersected with $\mathrm{Grass}(k,H)$ that is neither empty nor the whole Grassmannian. A proof of (1) also settles (2). Part (3) asks for a geometric interpretation, in the geometry of the Teichmüller flow, that attaches the eigenvalue-1 Jordan block of the concatenated Kontsevich-Zorich matrices to an explicitly identified class of Teichmüller-flow orbits (or of degenerations of translation surfaces). (1) General version (Problem 4.13 of the source): if $\mathcal{B}$ acts minimally on $\mathbf{P}H$, is $\mathcal{B}$ necessarily twisting? (2) Symplectic version (the weaker question the authors single out): if $\dim H=2g$, $\mathcal{B}$ preserves a symplectic form $\omega$ on $H$, and $\mathcal{B}$ acts minimally on $\mathbf{P}H$, does $\mathcal{B}$ twist every isotropic subspace $F$ (every subspace with $\omega|_F=0$ and $1\le\dim F\le g$) in the sense above? The logical structure is part of the question: an affirmative answer to (1) implies (2), while a counterexample to (1) need not be symplectic, so (2) is genuinely weaker; both are asked for arbitrary monoids of automorphisms, not only those arising from renormalization. For orientation: Avila-Viana's own route from projective minimality to twisting passes through minimality on the space of Lagrangian flags and uses an additional stabilizer hypothesis (their Lemma 2.7), and for the concrete Rauzy monoids twisting is now also derivable from the Zariski density of the generated group - so the question is precisely whether the topological hypothesis alone suffices. (3) Geometric component (Remark 6.11 of the source): give a geometric interpretation, in terms of the Teichmüller flow on the moduli space of translation surfaces, of how the parabolic behavior arises - the parabolic behavior that the Avila-Viana proof produces by concatenating Rauzy paths. The proof extracts this behavior from the parabolic block structure (an eigenvalue 1 with a nontrivial Jordan block in the central part) of the concatenated Kontsevich-Zorich matrices, which are used to prove strong pinching - a strengthened form of pinching, stated in the symplectic setting $\dim H=2g$ as in part (2), in which for every $C>0$ some $x\in\mathcal{B}$ has $\ln\sigma_g(x)>C$ and $\ln\sigma_i(x)>C\ln\sigma_{i+1}(x)$ for $1\le i\le g-1$, where the $\sigma_i(x)$ are the singular values of $x$ (Definition 4.5 in Section 4.2 of the source paper; strong pinching for the Rauzy monoids is proved in its Section 6, culminating in Theorem 6.1 via Lemma 6.10). The third question originally left open by the same source - the Zariski closure of the group generated by the Rauzy monoid (Remark 6.12; the implied Zariski density is Zorich's 1999 conjecture) - has been resolved in the literature and is not part of this problem; see Avila-Matheus-Yoccoz (ref. 3) and Gutiérrez-Romo (ref. 4).

References

1. Artur Avila and Marcelo Viana, 'Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture', Acta Mathematica 198 (2007), no. 1, 1-56. DOI: 10.1007/s11511-007-0012-1, https://doi.org/10.1007/s11511-007-0012-1 (preprint: arXiv:math/0508508, https://arxiv.org/abs/math/0508508)
2. Artur Avila and Marcelo Viana, 'Simplicity of Lyapunov spectra: a sufficient criterion', Portugaliae Mathematica 64 (2007), no. 3, 311-376. DOI: 10.4171/pm/1789, https://doi.org/10.4171/pm/1789 (preprint: arXiv:math/0607757, https://arxiv.org/abs/math/0607757)
3. Artur Avila, Carlos Matheus, and Jean-Christophe Yoccoz, 'Zorich conjecture for hyperelliptic Rauzy-Veech groups', Mathematische Annalen 370 (2018), no. 1-2, 785-809. DOI: 10.1007/s00208-017-1568-5, https://doi.org/10.1007/s00208-017-1568-5 (preprint: arXiv:1606.01227, https://arxiv.org/abs/1606.01227)
4. Rodolfo Gutiérrez-Romo, 'Classification of Rauzy-Veech groups: proof of the Zorich conjecture', Inventiones mathematicae 215 (2019), no. 3, 741-778. DOI: 10.1007/s00222-018-0836-7, https://doi.org/10.1007/s00222-018-0836-7 (preprint: arXiv:1706.04923, https://arxiv.org/abs/1706.04923)
5. Rodolfo Gutiérrez-Romo, 'Simplicity of the Lyapunov spectra of certain quadratic differentials', arXiv:1711.02006, https://arxiv.org/abs/1711.02006
6. Mark Bell, Vincent Delecroix, Vaibhav Gadre, Rodolfo Gutiérrez-Romo, and Saul Schleimer, 'Diagonal flow detects topology of strata', arXiv:2101.12197, https://arxiv.org/abs/2101.12197
7. Anton Zorich, 'How do the leaves of a closed 1-form wind around a surface?', in 'Pseudoperiodic topology', American Mathematical Society Translations, Series 2, vol. 197, American Mathematical Society, Providence, RI, 1999, pp. 135-178. DOI: 10.1090/trans2/197/05, https://doi.org/10.1090/trans2/197/05
8. Maxim Kontsevich and Anton Zorich, 'Connected components of the moduli spaces of Abelian differentials with prescribed singularities', Inventiones mathematicae 153 (2003), no. 3, 631-678. DOI: 10.1007/s00222-003-0303-x, https://doi.org/10.1007/s00222-003-0303-x
9. Yves Benoist, 'Propriétés asymptotiques des groupes linéaires', Geometric and Functional Analysis 7 (1997), no. 1, 1-47. DOI: 10.1007/pl00001613, https://doi.org/10.1007/pl00001613
10. Simion Filip, 'Zero Lyapunov exponents and monodromy of the Kontsevich-Zorich cocycle', Duke Mathematical Journal 166 (2017), no. 4, 657-706. DOI: 10.1215/00127094-3715806, https://doi.org/10.1215/00127094-3715806 (preprint: arXiv:1410.2129, https://arxiv.org/abs/1410.2129)
11. Giovanni Forni, 'Deviation of ergodic averages for area-preserving flows on surfaces of higher genus', Annals of Mathematics 155 (2002), no. 1, 1-103. DOI: 10.2307/3062150, https://doi.org/10.2307/3062150