Classification of Rauzy–Veech groups: proof of the Zorich conjecture

Abstract

A. We classify the Rauzy-Veech groups of all connected components of all strata of the moduli space of translation surfaces in absolute homology, showing, in particular, that they are commensurable to arithmetic lattices of symplectic groups. As a corollary, we prove a conjecture of Zorich about the Zariski-density of such groups. 1 ⊺ γ . In particular, if π ′ = π (that is, if γ is a cycle), one has that B γ (acting on row vectors) belongs to Sp(Ω π , Z). The Rauzy-Veech group of π is the group generated by matrices of this form:Definition 2.1. Let R be a Rauzy class and π ∈ R be a fixed vertex. We define the Rauzy-Veech group RV(π) of π as the set of matrices of the form B γ ∈ Sp(Ω π , Z) where γ is a cycle onR with endpoints at π. We will always consider the action of RV(π) on row vectors unless explicitly stated.

On Selberg’s eigenvalue conjecture for moduli spaces of abelian differentials

“…without any congruence aspect, was obtained by Avila, Gouëzel and Yoccoz in [AGY06]. In an earlier version of this manuscript, for certain types of components , Theorem 1.4 was conditional on a conjecture of Zorich [Zor99] that has since been proved by Gutiérrez-Romo [Gut19].…”

Counting saddle connections in a homology class modulo <inline-formula><tex-math id="M1">\begin{document}$ \boldsymbol q $\end{document}</tex-math></inline-formula> (with an appendix by Rodolfo Gutiérrez-Romo)

“…, 2m n ) spin , where spin ∈ {even, odd}, we take π ∈ {σ 2g , τ 2g } having the same spin parity as C . We can insert letters iteratively using the following two facts: genus-preserving simple extensions preserve the spin parity [12,Lemma 6.4], and singularities can be split in any way using simple extensions [12,Lemma 6.5]. To ensure that the lift of σ belongs to the Z-submodule generated by the sides associated with the new letters, we move along the strata in the following way (each arrow represents a simple extension):…”

“…Let Γ(q) be the kernel of the action of Γ on H 1 (Z q ). We start by adapting the classification of Rauzy-Veech groups [3,12] to our context. Indeed, for each permutation π there exist natural maps…”

“…We identify H 1 (S π , Z) = H 1 (Z) with the image of the second map. The Rauzy-Veech group induces a left action on H 1 (S π , Σ π , Z) = H rel 1 (Z) that preserves H σ 1 (Z) [12,Section 6]. The Rauzy-Veech group Ξ σ π,∞ is precisely the group of symplectic automorphisms induced by the restriction of this action to H σ 1 (Z).…”

Relative homological representations of framed mapping class groups

“…By deep results of Eskin, Filip, and Wright [4], the Zariski closure of is equal to (that is, it is ‘as big as possible’ given the constraint arising from the intersection pairing on absolute homology). The action of on absolute homology was determined by Gutierrez‐Romo [6], but explicit characterizations of the full group were only known for hyperelliptic components of strata [1, Corollary 2.8] and for the nonhyperelliptic components of (and ) [6, Theorem 5.1].…”