A new construction of subgroups of big mapping class groups

TL;DR

This work advances the understanding of subgroups of big mapping class groups by introducing shift maps and multipush maps to construct intrinsically infinite-type subgroups on uncountably many infinite-type surfaces. A key innovation is non-conjugate, non-surface-induced embeddings of indicable groups, including free groups, wreath products, and solvable Baumslag–Solitar groups, realized in Schreier surfaces built from graphs. The authors develop a robust combination theorem (a star-product framework) that assembles new indicable subgroups from existing ones, yielding embeddings not contained in the isometry group or in the closure of compactly supported maps. The results unify and extend prior constructions (e.g., RAAGs via Bestvina–Brady groups) and demonstrate broad applicability to infinite-type surfaces, significantly enriching the algebraic landscape of big mapping class groups and offering new avenues for geometric topology and group theory.

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.

Figures (15)

• Figure 1: A surface $S$ that admits a shift whose domain is an embedded copy of $D_\Pi$.
• Figure 2: An example of the surface $S_\Gamma(\Pi)$ where the graph $\Gamma$ is the Cayley graph of the group $\mathbb Z^2=\langle a,b : [a,b]\rangle$.
• Figure 3: Each graph can be realized as a Schreier graph but not a Cayley graph. The graph on the left has 3 ends, and the graph on the right has end space homeomorphic to the $2$-point compactification of $\mathbb Z$.
• Figure 4: A portion of the domain $D_a$ (in blue) of the multipush $x_a$ on the surface $S_\Gamma(\Pi)$, where $\Gamma$ is the Cayley graph $\Gamma=\Gamma(\mathbb F_2,\{a,b\})$ for $\mathbb F_2=\langle a,b\rangle$.
• Figure 5: A multipush on a surface corresponding to the Schreier graph on the left in \ref{['fig:SchreierExs']} that contains both a finite and infinite push. The domain of the multipush is highlighted in orange.

Theorems & Definitions (42)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Theorem 2.2: Classification of infinite-type surfaces
• Definition 2.3
• Lemma 2.4: ACCL20
• Definition 2.5
• Remark 2.6
• Definition 2.7