Topological normal generation of big mapping class groups

TL;DR

The paper investigates topological normal generation for big mapping class groups of tame infinite-type surfaces under CB-generation, showing a sharp dichotomy: when $E(S)$ is countable, $\mathrm{Map}(S)$ is topologically normally generated exactly when $S$ is uniquely self-similar; for uncountable $E(S)$ it provides a sufficient construction and a semidirect product description of $\mathrm{FMap}(S)$, enabling explicit generator bounds. It develops cohomological obstructions via $H^1(\mathrm{FMap}(S),\mathbb{Z})$ built from generalized shifts, and uses these to control when a single normal generator can exist. The results yield quadratic growth in the minimal number of normal generators, determined by topology-dependent constants $M$ and $C$, and connect to the rank of the abelianization through Field–Patel–Rasmussen-type bounds and a detailed analysis of end-space structure. Together with a uncountable-end-space framework and swindle techniques, the work provides a broad suite of examples and methods for understanding Rokhlin-type properties in big mapping class groups and clarifies when normal generation can or cannot occur in these rich groups.

Abstract

A topological group $G$ is \emph{topologically normally generated} if there exists $g \in G$ such that the normal closure of $g$ is dense in $G$. Let $S$ be a tame, infinite type surface whose mapping class group $\mathrm{Map}(S)$ is generated by a coarsely bounded set (CB generated). We prove that if the end space of $S$ is countable, then $\mathrm{Map}(S)$ is topologically normally generated if and only if $S$ is uniquely self-similar. Moreover, when the end space of $S$ is uncountable, we provide a sufficient condition under which $\mathrm{Map}(S)$ is topologically normally generated. As a consequence, we construct uncountably many examples of surfaces that are not telescoping yet have topologically normally generated mapping class groups. Additionally, we establish the semidirect product structure of $\mathrm{FMap}(S)$, the subgroup of $\mathrm{Map}(S)$ that pointwisely fixes all maximal ends that each is isolated in the set of maximal ends of $S$. This result leads to a proof that the minimum number of topological normal generators of $\mathrm{Map}(S)$ is bounded both above and below by constants that depend only on the topology of $S$. Furthermore, we demonstrate that the upper bound grows quadratically with respect to this constant.

Figures (9)

• Figure 1: Each maximal end is distinct from the others. A separating curve on the left is mapped to the right one by handle shift twice and puncture shift three times, both shifts toward to the right. If we assign the rightward direction as positive, this mapping corresponds to the value $(3,2)$.
• Figure 2: Biinfinite strip $\mathcal{B}$
• Figure 3: Embedded biinfinite strip $\mathcal{B}_z(A, B)$, where $z \sim \omega+1$ and both $A, B$ are $\omega^2+1$, and $(E(S), E^G(S)) = (\omega^2\cdot 2 + 1, \emptyset)$.
• Figure 4: Surface decomposition for countable end space. Here we have $K_1, \cdots, K_6$ of unique maximal end $x \sim \omega + 1$, $K_1'$ of the end accumulated by genus only, and $K_2'$ of end accumulated by both genus and punctures.
• Figure 5: Each $\sigma$(right axis) and $\tau$(left axis) is the $\pi$ rotation along the axis in the middle, respectively.

Theorems & Definitions (92)

• Theorem 1.1
• Theorem 1.2
• Lemma 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Corollary 1.8
• Definition 2.1
• Definition 2.2: Preorder on $E(S)$, mann2023large