Boundary of equisymmetric loci of Riemann surfaces with abelian symmetry

TL;DR

This paper investigates the boundary structure of equisymmetric loci in the moduli space of Riemann surfaces with abelian automorphism groups, describing the topological types of boundary strata, especially for hyperelliptic and cyclic p-gonal cases.

Contribution

It characterizes the boundary of equisymmetric loci with abelian symmetry in the moduli space, including explicit descriptions for hyperelliptic and cyclic p-gonal actions.

Findings

Determined the topological type of maximal boundary strata for abelian automorphism groups.

Described boundary strata in hyperelliptic and cyclic p-gonal cases using trees with fixed edges.

Provided a detailed stratification of the boundary in terms of topological types.

Abstract

Let ${\mathcal M}_g$ be the moduli space of compact connected Riemann surfaces of genus $g\geq 2$ and let $\widehat{{\mathcal M}_g}$ be its Deligne-Mumford compactification, which is stratified by the topological type of the stable Riemann surfaces. We consider the equisymmetric loci in $\mathcal M_g$ corresponding to Riemann surfaces whose automorphism group is abelian and determine the topological type of the maximal dimension strata at their boundary. For the particular cases of the hyperelliptic and the cyclic $p$-gonal actions, we describe all the topological strata at the boundary in terms of trees with a fixed number of edges.