We describe the Kontsevich-Zorich cocycle over an affine invariant orbifold coming from a (cyclic) covering construction inspired by works of Veech and McMullen. In particular, using the terminology in a recent paper of Filip, we show that all cases of Kontsevich-Zorich monodromies of SU (p, q) type are realized by appropriate covering constructions. The Veech-McMullen family F 1, for odd, resp. even, is the hyperelliptic component of the stratum H(( − 2 − )/ , . . . , ( − 2 − )/ ) of

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... n surfaces with conical singularities with total angle 2πk( −2) ; see [5] . In general, the Veech-McMullen family F k, is an affine suborbifold of the stratum H(k − 1, . . . , k − 1 , k( − 2 − )/ , . . . , k( − 2 − )/ ) given by a covering construction. Therefore, the Kontsevich-Zorich cocycle over the Teichmüller geodesic flow on F k, is coded by the hyperelliptic Rauzy diagrams with arrows decorated by certain matrices (describing actions on the homology of canonical translation surfaces in F k,l ). In particular, following the discussion in our previous paper [1], one can associate a Rauzy-Veech group RV (k, ) to F k, . By definition, the matrices in RV (k, ) preserve the natural symplectic intersection form on the absolute homology of the translation surfaces in F k, . REMARK 1.4. In this notation, the main result from our previous paper [1] asserts that RV (1, k) is naturally isomorphic to an explicit finite-index subgroup of the integral symplectic group Sp(2g , Z) (where g is the genus of M k, ). Statement of the main result. In this paper, we study the structure of the Rauzy-Veech groups of F k, . THEOREM 1.5. The real Hodge bundle over F k, decomposes into a direct sum H 1 ⊕ · · · ⊕ H k of flat subbundles H r associated to the eigenspaces of the generator of the deck group of M k, → M 1, (cf. (2) and (3) below). The Rauzy-Veech group of F k, respects this decomposition. Moreover, if one denotes by RV (k, )| H r the group associated to the restrictions to H r of the matrices in RV (k, ), then: (a) RV (k, )| H k is naturally isomorphic to RV (1, ); thus, RV (k, )| H k is isomorphic to a finite-index subgroup of Sp(2g , Z); (b) for each 0 < r < k, the symplectic intersection form on H r induces a Hermitian form Q r /2k of signature ( (r /2k) − 1 , (1 − (r /2k)) − 1 ) which is preserved by RV (k, )| H r ; furthermore, if (r /2k) < 1 and r /2k = 1/6, 1/4, then RV (k, )| H r ∩SU (Q r /2k ) is dense in SU (Q r /2k ) (for the usual topology); if (r /2k) ∉ Z and r /2k = 1/6, 1/4, 1/3, then RV (k, )| H r ∩ SU (Q r /2k ) is Zariski dense in SU (Q r /2k ). REMARK 1.6. Actually, our discussion in Section 4 provides a precise version of Theorem 1.5: in particular, we compute the Zariski closure of RV ( , k)| H r ∩ SU (Q r /2k ) in the exceptional cases r /2k = 1/6, 1/4, 1/3. However, we have not included all possibilities in Theorem 1.5 in order to get a "cleaner" statement. This result provides explicit examples showing that all cases of SU (p, q) Kontsevich-Zorich monodromies discussed in Filip's paper [3] actually occur. A direct consequence of Theorem 1.5 and the simplicity criterion of Avila-Viana [2] (as stated in Subsection 2.5 of [6]) is the following. JOURNAL OF MODERN DYNAMICS VOLUME 14, 2019, 21-54 4.5.3. Application. PROPOSITION 4.6. Assume that d ≥ 3, ρ d , ρ d +1 = 1. Assume also that the intersection of the group generated by the operators L q , q ∈ A d −1 , on C A d −1 with the special unitary group SU (Q α ) is dense (resp. Zariski dense) in SU (Q α ). Then the intersection of the group generated by the operators L p , p ∈ A d , on C A d with the special unitary group SU (Q α ) is dense (resp. Zariski dense) in SU (Q α ) . Proof. Let p 1 , p 2 be two distinct elements of A d . Denote by u 1 , u 2 generators of H p 1 , H p 2 respectively, satisfying Q α (u 1 ) = Q α (u 2 ) = 0. Such a choice is possible because the restrictions of Q α to H p 1 , H p 2 have the same signature by lemma 4.2.