Teichmüller discs with completely degenerate Kontsevich–Zorich spectrum

Abstract

Title of dissertation: PROGRESS TOWARD CLASSIFYING TEICHMULLER DISKS WITH COMPLETELY DEGENERATE KONTSEVICH-ZORICH SPECTRUM David Aulicino, Doctor of Philosophy, 2012 Dissertation directed by: Professor Giovanni Forni Department of Mathematics We present results toward resolving a question posed by Eskin-KontsevichZorich and Forni-Matheus-Zorich. They asked for a classification of all SL2(R)invariant ergodic probability measures with completely degenerate Kontsevich Zorich spectrum. Let Dg(1) be the subset of the moduli space of Abelian differentials Mg whose elements have period matrix derivative of rank one. There is an SL2(R)-invariant ergodic probability measure ν with completely degenerate Kontsevich-Zorich spectrum, i.e. λ1 = 1 > λ2 = · · · = λg = 0, if and only if ν has support contained in Dg(1). We approach this problem by studying Teichmuller disks contained in Dg(1). We show that if (X,ω) generates a Teichmuller disk in Dg(1), then (X,ω) is completely periodic. Furthermore, we show that there are no Teichmuller disks in Dg(1), for g = 2, and the known example of a Teichmuller disk in D3(1) is the only one. Finally, we present an idea that might be able to fully resolve the problem. PROGRESS TOWARD CLASSIFYING TEICHMULLER DISKS WITH COMPLETELY DEGENERATE KONTSEVICH-ZORICH SPECTRUM