Monodromy kernels for strata of translation surfaces

TL;DR

This paper investigates the topological monodromy of certain translation surface strata, showing it does not support the conjectured commensurability with mapping class groups by analyzing specific genus 3 components.

Contribution

It provides new results on the monodromy maps for specific strata, demonstrating their kernels contain large free groups, thus challenging existing conjectures.

Findings

Monodromy maps for H(3,1) and H^{nh}(4) are not commensurable with mapping class groups.

Kernels of these monodromy maps contain non-abelian free groups of rank 2.

Results build on and extend work by Wajnryb.

Abstract

The non-hyperelliptic connected components of the strata of translation surfaces are conjectured to be orbifold classifying spaces for some groups commensurable to some mapping class groups. The topological monodromy map of the non-hyperelliptic components projects naturally to the mapping class group of the underlying punctured surface and is an obvious candidate to test commensurability. In the present article, we prove that for the components $\mathcal{H}(3,1)$ and $\mathcal{H}^{nh}(4)$ in genus 3 the monodromy map fails to demonstrate the conjectured commensurability. In particular, building on work of Wajnryb, we prove that the kernels of the monodromy maps for $\mathcal{H}(3,1)$ and $\mathcal{H}^{nh}(4)$ are large, as they contain a non-abelian free group of rank $2$