We explicitly construct new subgroups of the mapping class groups of an uncountable collection of infinite-type surfaces, including, but not limited to, free groups, Baumslag-Solitar groups, mapping class groups of other surfaces, and a large collection of wreath products. For each such subgroup H and surface S, we show that there are countably many non-conjugate embeddings of H into map(S); in certain cases, there are uncountably many such embeddings. The images of each of these embeddings cannot lie in the isometry group of S for any hyperbolic metric and are not contained in the closure of the compactly supported subgroup of map(S). In this sense, our construction is new and does not rely on previously known techniques for constructing subgroups of mapping class groups. Notably, our embeddings of map(S') into map(S) are not induced by embeddings of S' into S. Our main tool for all of these constructions is the utilization of special homeomorphisms of S called shift maps, and more generally, multipush maps.

The authors would like to thank Women in Groups, Geometry, and Dynamics (WiGGD) for facilitating this collaboration, which was supported by NSF DMS--1552234, DMS--1651963, and DMS--1848346. The authors also thank Mladen Bestvina and Robbie Lyman for helpful conversations, as well as George Domat for productive discussions about surfaces with indicable mapping class groups and an anonymous referee providing comments which significantly improved the paper. In addition, the authors acknowledge support from NSF grants DMS--1803368, DMS--2106906, and DMS-2340341 (Abbott), DMS--1906095 and RTG DMS--1840190 (Hoganson), DMS--1902729 and DMS--2231286 (Loving), DMS--1937969 and DMS--2046889 (Patel), and DMS--2005297 and DMS--2506840 (Skipper). Skipper was also supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No.725773).