Posted by: matheuscmss | June 16, 2017

Zorich’s conjecture on Zariski density of Rauzy-Veech groups (after Gutiérrez-Romo)

Rodolfo Gutiérrez-Romo has just uploaded to the arXiv his preprint Zariski density of Rauzy–Veech groups: proof of the Zorich conjecture.

This article is part of the PhD thesis project of Rodolfo (under the supervision of Anton Zorich and myself), which started last September 2016. (In fact, one of my motivations to obtain a “Habilitation à Diriger des Recherches” degree last June 2, 2017 was precisely to be able to formally co-supervise Rodolfo’s PhD thesis project.)

In this (short) blog post, we discuss some aspects of Rodolfo’s solution to Zorich conjecture (and we refer to the preprint for the details).

1. Statement of Zorich conjecture

The study of Lyapunov exponents of the Kontsevich-Zorich cocycle (and, more generally, variations of Hodge structures) found many applications since the pioneer works of Zorich and Forni in the late nineties:

  • Zorich and Forni described the deviations of ergodic averages of typical interval exchange maps and translation flows in terms of Lyapunov exponents;
  • Avila and Forni used in 2007 the positivity of second Lyapunov of the Kontsevich-Zorich cocycle with respect to Masur-Veech measures (among many other ingredients) to show that typical interval exchange transformations and translation flows are weak mixing;
  • Delecroix, Hubert and Lelièvre confirmed in 2014 a conjecture of Hardy and Weber on the abnormal rate of diffusion of typical trajectories on {\mathbb{Z}^2}-periodic Ehrenfest wind-tree models of Lorenz gases;
  • Kappes and Möller completed in 2016 the classification of commensurability classes of non-arithmetic lattices of {PU(1,n)}, {n\geq 2}, constructed by Deligne and Mostow in the eighties thanks to new invariants coming from Lyapunov exponents;
  • etc.

The success of Zorich in describing such deviations of ergodic averages together with many numerical experiments led Kontsevich and him to conjecture that the Lyapunov exponents of the Kontsevich-Zorich (KZ) cocycle with respect to Masur-Veech measures are simple (i.e., their multiplicities are {1}).

Moreover, Zorich had in mind a specific way to establish the Kontsevich-Zorich conjecture: first, he conjectured that the so-called Rauzy-Veech groups (associated to a certain combinatorial description of the matrices of the KZ cocycle appearing along typical trajectories for the Masur-Veech measures) are Zariski-dense in the symplectic groups {Sp(2g,\mathbb{R})}, {g\geq 1}; then, he noticed that the works of Guivarc’h-Raugi and Goldsheid-Margulis on the simplicity of Lyapunov exponents for random products of matrices forming a Zariski-dense subgroup could be useful to deduce the “Kontsevich-Zorich simplicity conjecture” from his “Zariski density conjecture”.

After an important partial result of Forni in 2002, Avila and Viana famously established the Kontsevich-Zorich conjecture in 2007. Nevertheless, the arguments of Avila and Viana were slightly different from the scheme outline by Zorich: indeed, as they pointed out in Remark 6.12 of their paper, Avila and Viana avoided discussing the Zariski closure of Rauzy-Veech groups by showing that Rauzy-Veech groups are pinching and twisting, and that these two properties suffice to get the simplicity of the Lyapunov spectrum (i.e., Kontsevich-Zorich conjecture).

Remark 1 It is worth to notice that Zariski density implies pinching and twisting, but the converse is not true in general.

In summary, the solution of the Kontsevich-Zorich conjecture by Avila and Viana via the pinching and twisting properties for Rauzy-Veech groups left open Zorich’s conjecture on the Zariski density of Rauzy-Veech groups.

Remark 2 Besides giving stronger information about Rauzy-Veech groups (and, in particular, a new proof of Avila and Viana theorem), Zorich’s conjecture has other applications: for example, Magee recently showed that the validity of Zorich’s conjecture implies that the spectral gap / rate of mixing of the geodesic flow on congruence covers of connected components of the strata of moduli spaces of unit area translation surfaces is uniform.

2. Hyperelliptic Rauzy-Veech groups

As we already discussed in this blog, Avila, Yoccoz and myself were able to prove Zorich’s conjecture in the particular case of hyperelliptic Rauzy-Veech groups by showing the stronger statement that such groups contain an explicit finite-index subgroup of {Sp(2g,\mathbb{Z})}: roughly speaking, the Rauzy-Veech group is the subgroup of {Sp(2g,\mathbb{Z})} consisting of matrices whose reduction modulo two permute the basis vectors {e_1,\dots, e_{2g}} and {\sum\limits_{k=1}^{2g} e_k}.

As it turns out, the hyperelliptic Rauzy-Veech groups are associated to one of the three connected components of the so-called minimal strata {\mathcal{H}(2g-2)} (consisting of translation surfaces of genus {g} with a unique conical singularity of total angle {2\pi(2g-1)}): in fact, it was proved by Kontsevich and Zorich in 2003 that the minimal strata (of genus {g\geq 4}) have three connected components called hyperelliptic {\mathcal{H}(2g-2)^{hyp}}, even spin {\mathcal{H}(2g-2)^{even}} and odd spin {\mathcal{H}(2g-2)^{odd}}.

3. Rauzy-Veech groups of minimal strata

As a warm-up problem, we asked Rodolfo to perform numerical experiments with the Rauzy-Veech groups of the odd connected component {\mathcal{H}(4)^{odd}} of the minimal stratum in genus 3 and the even and odd connected components of the minimal stratum in genus 4. In particular, we told him to “compute” the indices of the reductions modulo two of such a Rauzy-Veech group in {Sp(6,\mathbb{Z}/2\mathbb{Z})} and {Sp(8,\mathbb{Z}/2\mathbb{Z})}.

After playing a bit with the matrices in his computer, Rodolfo announced (among many other things) that the index in {Sp(6,\mathbb{Z}/2\mathbb{Z})} of the Rauzy-Veech group of {\mathcal{H}(4)^{odd}} was 28.

This number ringed a bell because (as it is briefly explained here for instance) {Sp(2g,\mathbb{F}_2)} contains two orthogonal subgroups {O^{even}}, resp. {O^{odd}}, of indices {2^{g-1}(2^g+1)}, resp. {2^{g-1}(2^g-1)}, consisting of matrices stabilizing a quadratic form with even, resp. odd Arf invariant of representing the reduction modulo two of the symplectic form. In particular, the fact that the number {28 = 2^{3-1}(2^3-1)} matches the index of {O^{odd}} suggest the conjecture that Rauzy-Veech groups of {\mathcal{H}(2g-2)^{odd}}, resp. {\mathcal{H}(2g-2)^{even}}, is the pre-image of {O^{odd}}, resp. {O^{even}} in {Sp(2g,\mathbb{Z})} under the reduction modulo two map {Sp(2g,\mathbb{Z})\rightarrow Sp(2g,\mathbb{Z}/2\mathbb{Z})}.

Once we convinced ourselves about the plausibility of this conjecture, Rodolfo started working on the geometry of the corresponding Rauzy diagrams (graphs underlying the structure of the Rauzy-Veech groups) in order to figure out a systematic way of producing many particular matrices generating the desired candidate groups above.

As it turns out, Rodolfo did this in two steps (which occupy most [15 pages] of his preprint):

  • first, he exploits the fact that the level two congruence subgroup {\Gamma_2(2g)} of {Sp(2g,\mathbb{Z})} (i.e., the kernel of the natural map {Sp(2g,\mathbb{Z})\rightarrow Sp(2g,\mathbb{Z}/2\mathbb{Z})}) is generated by the squares of certain symplectic transvections to show that the Rauzy-Veech groups of the odd and even components of {\mathcal{H}(2g-2)} contain {\Gamma_2(2g)}; for this sake, he exhibits a rich set of loops in Rauzy diagrams inducing appropriate Dehn twists (giving “most” of the desired symplectic transvections).
  • secondly, he proves that the reduction modulo two of the Rauzy-Veech groups of the odd, resp. even components of {\mathcal{H}(2g-2)} coincides with {O^{odd}}, resp. {O^{even}}, by using the fact that {O^{odd}} and {O^{even}} are generated by orthogonal transvections.

In summary, Rodolfo showed that the Rauzy-Veech groups of the odd and even components of {\mathcal{H}(2g-2)} are explicit subgroups of {Sp(2g,\mathbb{Z})} of indices {2^{g-1}(2^g-1)} and {2^{g-1}(2^g+1)}.

By putting this result together with the result by Avila, Yoccoz and myself for hyperelliptic Rauzy-Veech groups, we conclude that the Rauzy-Veech group of any connected component of a minimal stratum {\mathcal{H}(2g-2)} is a finite-index subgroup of {Sp(2g,\mathbb{Z})} and, a fortiori, a Zariski-dense subgroup of {Sp(2g,\mathbb{R})}.

4. Rauzy-Veech groups of general strata

Philosophically speaking, a general translation surface {X} of genus {g} “differs” from a translation surface {Y} in the minimal stratum {\mathcal{H}(2g-2)} because of the relative homology produced by the presence of many conical singularities. In particular, if we “merge” conical singularities of {X}, we should find a translation surface {Y\in\mathcal{H}(2g-2)}.

From the geometrical point of view, this philosophy was made rigorous by Kontsevich and Zorich in 2003: indeed, they formalized the notions of “merging” and “breaking” zeroes in order to “reduce” the classification of connected components of general strata to the case of connected components of minimal strata!

From the combinatorial point of view, Avila and Viana obtained the following combinatorial analog of Kontsevich-Zorich geometrical statement: we can merge conical singularities until we end up with a component of a minimal stratum in such a way that a “copy” of the Rauzy-Veech group of a component of a minimal stratum shows up inside any Rauzy-Veech group.

Hence, this result of Avila-Viana (or rather its variant stated as Lemma 6.3 in Rodolfo’s preprint) allows Rodolfo to conclude his proof of (a statement slightly stronger than) Zorich’s conjecture: the Rauzy-Veech group of any connected component of any stratum of the moduli space of unit area translation surfaces of genus {g} is a finite-index subgroup of {Sp(2g,\mathbb{Z})} simply because the same is true for the connected components of the minimal stratum {\mathcal{H}(2g-2)}.


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