Representation stability for moduli spaces of admissible covers

Abstract

We prove a representation stability result for the sequence of spaces ${\overline{M}}_{g,n}^A$ of pointed admissible $A$-covers of stable $n$-pointed genus-$g$ curves, for an abelian group $A$. For fixed genus $g$ and homology degree $i$, we give the sequence of rational homology groups $H_i({\overline{M}}_{g,n}^A;\mathbb{Q})$ the structure of a module over a combinatorial category, a la Sam--Snowden, and prove that this module is generated in degree at most $g+5i$. This implies that the generating function for the ranks of the homology groups is rational, with poles in the set $\left\{-1,-\frac{1}{2},\dots ,-\frac{1}{|A{|}^2\cdot (g+5i)}\right\}$. In the case where $A$ is the trivial group, our work significantly improves on previous representation stability results on the Deligne--Mumford compactification ${\overline{M}}_{g,n}$.

arXiv comment26 pages