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Let $S_{g,p}^n$ be a compact oriented surface of genus $g$ with $p$ punctures and $n$ boundary components, and $\operatorname{Mod}(S)$ and $\operatorname{PMod}(S)$ be the mapping class group and the p …

We have the following pushout diagram: $$\begin{array}{ccc} \langle X, Y \rangle & \xrightarrow{\alpha} & \mathbb{Z}_a \ast \mathbb{Z}_b \ast \mathbb{Z}_c \ast \mathbb{Z}_d \\ \downarrow \scriptsize{\ …

For a group $G$, there are two elements a, b which are "relation-free", i.e., there is no nonempty, reduced word $W(X,Y)$ such that $W(a,b)=1$ in $G$. Is there any terminologies or theories for that? …

For two subgroups $A, B$ in $G$, $[A,A] \cap [B,B] = [A\cap B, A \cap B]$? At least, if $G$ is free, is the left contained in the right?

Let $F$ be a free group $\langle x_1,...x_m\rangle$. If $a\in F_2$ and $[a,x_1] \in F_n$ then $a\in F_{n-1}$. Here, $F_n$ is the $n$-th lower central series term with $F_2=[F:F]$. How can I prove t …

I have a question about some quotient groups of the lower central series of a free group. When there's a free group $F = \langle x_1,\cdots, x_n, y_1, \cdots, y_m\rangle $, let $A$ be the subgroup g …

Hello? I have some questions in the group theory. I know that the intersection of the lower central series of a finitely generate free group is trivial. So I wonder whether every nontrivial subgroup o …

Hello? I have a simple question about combinatorial group theory. For a group $G$, I saw $[G_k, G_m] \subset G_{k+m}$ and these two subgroups need not be equal. Then is there any known condition that …