4. Algebraic and topological properties of big mapping class groups

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? 
Problem 4.1.
For every $S$ does there exist a countable group $G$ such that $G$ is not a subgroup of $\operatorname{Map}(S)$? 
Problem 4.15.
Describe the compact subsets/subgroups of $\operatorname{Map}(S)$. Also describe all bounded subsets/subgroups of $\operatorname{Map}(S)$. 
Problem 4.25.
Can we see the topology of $S$ in the algebraic structure of $\operatorname{Map}(S)$? 
Problem 4.35.
Are there any finite nonabelian 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 finitetype surfaces. Those finitetype 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 quasiisometric? 
Problem 4.5.
Is every injective selfhomomorphism from $\operatorname{Map}(S)$ an isomorphism? That is to say, for which $S$ is $\operatorname{Map}(S)$ coHopfian (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 higherrank lattices map into these groups? 
Problem 4.6.
For surfaces for which this is not known, can we produce quasimorphisms using actions of $\operatorname{Map}(S)$? 
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 $\mathbb{R}^2C$, is it true that every subgroup of $\operatorname{Map}(S)$ has either infinitedimensional 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. 
Problem 4.8.
For any homomorphism from $\operatorname{Map}(S)$ to a separable topological group $G$, is this automatically continuous?
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.