Simplicity of the Lyapunov spectra of certain quadratic differentials

Abstract

A. We prove that the "plus" Rauzy-Veech groups of all connected components of the strata of meromorphic quadratic di erentials de ned on Riemann surfaces of genus at least one having at most simple poles and at least three singularities (zeros or poles), not all of even order, are nite-index subgroups of their ambient symplectic groups. This shows that the "plus" Lyapunov spectrum of such strata is simple. Moreover, we show that the index of the "minus" Rauzy-Veech group is also nite for connected components of strata satisfying the same conditions and having exactly two singularities of odd order. This shows that the "minus" Lyapunov spectrum of such strata is simple.

The flow group of rooted abelian or quadratic differentials

“…The notion of simple extensions was original introduced by Avila-Viana [AV07b] in their proof of the Kontsevich-Zorich conjecture for abelian differentials. By expanding it to type changing extensions, the notion was extended to quadratic differentials by the fourth author [Gut17].…”

“…See the work by Avila-Matheus-Yoccoz [AMY18] for hyperelliptic abelian components and by the fourth author [Gut19] for general abelian components. Plus and minus Rauzy-Veech groups for quadratic stratum components that can arise by a string of simple extensions from abelian base cases are also more tractable even though here a complete classification is not yet achieved [Gut17].…”

“…For the abelian case, this conjecture was established in the famous work by Avila-Viana [AV07b]. The quadratic case is known for many stratum components but not in full generality [Tre13;Gut17].…”

“…The groups of Theorem 1.4 that arise, by splitting singularities, from abelian strata are known to be finite index inside the ambient symplectic groups (over Z) and hence Zariski dense. This was done by Avila-Matheus-Yoccoz [AMY18] for abelian hyperelliptic components and by the fourth author [Gut19;Gut17] for all other components mentioned above. Using our techniques, and again replying on certain machinery from previous work, our Theorem 1.4 gives Zariski density in all cases.…”

“…As mentioned before, simplicity was known for all abelian [AV07b] and some quadratic stratum components [Gut17]. It is also known for the principal stratum of quadratic differentials by different methods through the recently announced solution by Eskin-Mirzakhani-Rafi of the Furstenberg problem for random walks on the mapping class group.…”