L-cut splitting of translation surfaces and non-embedding of pseudo-Anosovs (in genus two)
Volume 153 / 2018
Colloquium Mathematicum 153 (2018), 51-79
MSC: Primary 30F60, 37E30, 37D20, 37D50; Secondary 57R30, 55M20.
DOI: 10.4064/cm6956-6-2017
Published online: 9 April 2018
Abstract
We introduce a concept of a pair of parallel L-cuts on a translation surface, conjecture existence of such pairs for surfaces of genus $g \gt 1$, and find them for $g=2$. We discuss applications to genus reducing decomposition of surfaces and to pseudo-Anosov maps (concerning their abelian-Nielsen equivalence classes and non-embedding into toral automorphisms). In particular, we provide a negative answer to the question about injectivity of the Abel–Franks map for genus 2 pseudo-Anosovs with orientable foliations.
