Birman-Hilden theory for big mapping class groups

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.