About mapping class group.

Question

What is best book for self learning mapping class group?

I read "A Primer on Mapping Class Groups" By Benson Farb, Dan Margalit.

Is there a topological space $X$ where we don't know $\mathrm{MCG}(X)$ ?

I want to find some open problem in mapping class group .

Answer 1

The "Primer" is a good source (and so is Ivanov's book) but a bit old. A lot of things are known now that are not there and there are no books covering this information.

As for spaces $X$ such that $MCG(X)$ are not known, relatively little is known about the mapping class groups of 3-manifolds, although see this paper.

There are two famous open problems about MCG of surfaces: whether they can have Kazdan property (T) and whether they can contain surface subgroups consisting of pseudo-Anosov elements.

Answer 2

Here are some open problems on mapping class groups.

Question 1: Let $S_g$ denote the closed orientable surface of genus $g$. Is $\text{Mod}(S_g)$ linear?

Bigelow–Budney and Korkmaz were able to prove that $\text{Mod}(S_2)$ is linear. For $g\ge 3$ the conjecture is wide open.

Question 2: Is the $k$-th Lawrence representation of $\text{Mod}(S_g)$ faithful for any $k ≥ 1$?

Question 3: Is it true that every finite-index subgroup in $\text{Mod}(S_g)$ contains a congruence subgroup?