A new construction of subgroups of big mapping class groups

TL;DR

This paper introduces a novel method for explicitly constructing diverse subgroups within big mapping class groups of infinite-type surfaces, revealing new embeddings that are not derived from surface embeddings or isometry groups.

Contribution

The authors develop a new construction technique for subgroups of big mapping class groups using shift and multipush maps, expanding the known subgroup landscape.

Findings

Constructed new subgroups including free groups, Baumslag-Solitar groups, and wreath products.

Established existence of multiple non-conjugate embeddings for each subgroup and surface.

Demonstrated these embeddings are fundamentally different from isometry or surface embedding-based subgroups.

Abstract

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 $\textrm{Map}(S)$; 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 $\textrm{Map}(S)$. 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 $\textrm{Map}(S')$ into $\textrm{Map}(S)$ 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.