Affiliation:
1. School of Mathematics and Statistics , University of Glasgow, University Place, Glasgow G12 8QQ, UK
Abstract
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 the 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 $2$.
Publisher
Oxford University Press (OUP)
Reference42 articles.
1. Property ${P}_{naive}$ for acylindrically hyperbolic groups;Abbott;Math. Z.,2019
2. Parabolic subgroups acting on the additional length graph;Antolín;Algebr. Geom. Topol.,2021
3. Normal forms of functions near degenerate critical points, the Weyl groups ${A}_k$, ${D}_k$, ${E}_k$ and Lagrangian singularities;Arnol’d;Funct. Anal. Appl.,1972
4. Horocycle dynamics: new invariants and Eigenform loci in the stratum $\mathcal {H}$(1,1);Bainbridge;Mem. Amer. Math. Soc.,2022