  We explicitly construct new subgroups of the mapping class groups of an
uncountable collection of infinite-type surfaces, including, but not limited
to, right-angled Artin groups, 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 $\text{Map}(S)$; 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
$\text{Map}(S)$. 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 $\text{Map}(S')$ into $\text{Map}(S)$ 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.
