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## 4. Algebraic and topological properties of big mapping class groups

1. #### Problem 4.05.

Give an example of a pair $(S, G)$ of a surface and a countable group such that $G$ is not a subgroup of $\operatorname{Map}(S)$. What obstructions exist barring countable groups from being subgroups of families of big mapping class groups?
For example, $S^2-C$ is circularly orderable, so its subgroups must also be circularly orderable.
• #### Problem 4.1.

For every $S$ does there exist a countable group $G$ such that $G$ is not a subgroup of $\operatorname{Map}(S)$?
Aougab-Patel-Vlamis have shown that when $S$ is the Loch Ness monster surface, there is no such group $G$. What about other surfaces?
• #### Problem 4.15.

Describe the compact subsets/subgroups of $\operatorname{Map}(S)$. Also describe all bounded subsets/subgroups of $\operatorname{Map}(S)$.
• #### Problem 4.2.

Are there "nice" or "natural" generators for $\operatorname{Map}(S)$?
• #### Problem 4.25.

Can we see the topology of $S$ in the algebraic structure of $\operatorname{Map}(S)$?
• #### Problem 4.3.

Does $\operatorname{Map}(S)$ have a canonical word metric?
• #### Problem 4.35.

Are there any finite non-abelian quotients of $\operatorname{PMap}(S)$ or $\operatorname{Map}(S)$?
• #### Problem 4.4.

When $S$ is a finite genus surface, there are forgetful homomorphisms from $\operatorname{PMap}(S)$ to the mapping class groups of finite-type surfaces. Those finite-type mapping class groups are residually finite, so there are many further quotients. What other finite quotients can we have for $\operatorname{Map}(S)$ and $\operatorname{PMap}(S)$?
• #### Problem 4.45.

Let $S$ have infinite genus and no punctures, with finitely many ends accumulated by genus. Must every homomorphism factor through an abelian subgroup? Can you forget ends accumulated by genus? Are surfaces with infinite genus and no punctures, with finitely many ends accumulated by genus quasi-isometric?
• #### Problem 4.5.

Is every injective self-homomorphism from $\operatorname{Map}(S)$ an isomorphism? That is to say, for which $S$ is $\operatorname{Map}(S)$ co-Hopfian (or Hopfian)?
•     Lattices don’t map into $\operatorname{Mod}(S_g)$ for a compact surface $S_g$ of genus $g$.

#### Problem 4.55.

Do lattices map into $\operatorname{Map}(S)$ or $\overline {\operatorname{PMap}_c(S)}$? In particular, can higher-rank lattices map into these groups?
Every lattice maps into some $\operatorname{Map}(S)$.
• #### Problem 4.6.

For surfaces for which this is not known, can we produce quasimorphisms using actions of $\operatorname{Map}(S)$?
The answer is yes for $\mathbb{R}^2-C$ acting on the ray graph, or the loop graph, and for surfaces with a non-zero finite number of isolated punctures.
•     No big mapping class groups satisfies the strong Tits alternative, but it is not known about the classical Tits Alternatives.

#### Problem 4.65.

Are there any big mapping class groups that satisfy the classical Tits alternative?
• #### Problem 4.7.

What about bounded cohomology? In $\mathbb{R}^2-C$, is it true that every subgroup of $\operatorname{Map}(S)$ has either infinite-dimensional space of quasimorphisms or is amenable?
• #### Problem 4.75.

Given a homomorphism $f: \operatorname{Map}(S) \to \operatorname{Map}(S')$, does $f$ preserve the notion of being compactly supported? That is, does $f$ send compactly supported elements to compactly supported elements.
The answer is yes for surjective homomorphisms due to the characterization of compactly supported elements by Bavard-Dowdall-Rafi, but is not known in general.
• #### Problem 4.8.

For any homomorphism from $\operatorname{Map}(S)$ to a separable topological group $G$, is this automatically continuous?
A potential starting point is to consider the case when $G=\operatorname{Map}(S')$.
1. Remark. If $M$ is a compact manifold then $\operatorname{Homeo}(M)$ has this property.

Cite this as: AimPL: Surfaces of infinite type, available at http://aimpl.org/genusinfinity.