I will report on a joint project, in progress, with Fabrizio Catanese and Michael Lönne, where we investigate the homological stability of moduli spaces of (smooth) curves C with a given group of symmetry G, when the genus of the curves increases. Over the field of rational numbers, the homology of these moduli spaces can be interpreted as equivariant homology of the mapping class group Map_{g',d} acting on HV(G;g',d)/G, where HV(G;g',d) is the set  of Hurwitz vectors corresponding to coverings C->C/G, g' is the genus of C/G, d is the number of branch points of the covering C -> C/G, and G acts on HV(G;g',d) by conjugation. In the case where d=0, we prove that the equivariant homology groups of Map_{g'}^1 (the mapping class group of a surface of genus g' with one boundary component) acting on  HV(G;g',0) don't depend on g', for g' large enough. This is an intermediate step towards the study of homological stability of Map_{g'} acting on HV(G;g',0)/G.

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