Birman-Hilden theory for big mapping class groups

TL;DR

This paper extends Birman-Hilden theory to infinite-type surfaces and branched covers, showing that certain mapping class groups can be embedded into those of their orientable double covers, broadening the understanding of infinite surface symmetries.

Contribution

It generalizes Birman-Hilden theory to infinite-type surfaces and branched covers, establishing new embeddings of non-orientable surface groups into orientable ones.

Findings

Proves the Birman-Hilden property for fully ramified branched covers of infinite-type surfaces.

Shows that mapping class groups of non-orientable infinite-type surfaces embed into those of their orientable double covers.

Extends known results to the context of infinite degree branched coverings.

Abstract

Let $S$ and $X$ be two connected topological surfaces without boundary, and assume that $S$ is either of infinite type or has negative Euler characteristic. In this paper, we prove that if $p:S\rightarrow X$ is a fully ramified branched covering map, then $p$ satisfies the Birman-Hilden property. This generalizes a theorem of Winarski, and the known results in the literature, to the context of surfaces of infinite type and branched covering maps of infinite degree. As an application, we show that the mapping class group (respectively, the braid group on $k$-strands) of a non-orientable surface of infinite type can be realized as a subgroup of the mapping class group (respectively, the braid group on $2k$-strands) of its orientable double cover.