injectivity of pushout?

Question

We have the following pushout diagram: $$\begin{array}{ccc} \langle X, Y \rangle & \xrightarrow{\alpha} & \mathbb{Z}_a \ast \mathbb{Z}_b \ast \mathbb{Z}_c \ast \mathbb{Z}_d \\ \downarrow \scriptsize{\beta} && \downarrow \scriptsize{g} \\ \mathbb{Z}_e & \xrightarrow{f} & G\end{array}$$ Suppose that $\alpha$ is injective, and $\beta$ maps $X,Y$ to $1\in \mathbb{Z}_e$. ($\mathbb{Z}_n$ means the cyclic group of order $n$.)

I wonder that in which cases, $f$ is injective. For any positive integers $a,b,c,d,e \geq 2$? Otherwise, only for relative prime or distinct prime $a,b,c,d,e$?

(I found that if $\phi$ is injective, then $\psi$ is injective in "adhesive" categories. The category of abelian groups is adhesive, and the category of groups isn't. I also wonder which subcategory of groups are adhesive. ) $$\begin{array}{ccc}A & \xrightarrow{\phi} & B \\ \downarrow && \downarrow \\ C & \xrightarrow{\psi} & D\end{array}$$.

In the category of groups, there is a counterexample: http://math.stackexchange.com/questions/601463/a-monomorphism-of-groups-which-is-not-universal

I can change my diagram similarly to the above counterexample: $$\begin{array}{ccc}\mathbb{Z} & \xrightarrow{\alpha'} & (\mathbb{Z}_a \ast \mathbb{Z}_b \ast \mathbb{Z}_c \ast \mathbb{Z}_d)/\langle X=Y \rangle \\ \downarrow \scriptsize{\beta'} && \downarrow \scriptsize{g'} \\ \mathbb{Z}_e & \xrightarrow{f} & G\end{array}$$ $\alpha'(1)=X$, any words in $X,Y$ are nontrivial in $\mathbb{Z}_a \ast \mathbb{Z}_b \ast \mathbb{Z}_c \ast \mathbb{Z}_d$, and $\beta'$ is surjective. Then, $\alpha'$ is also injective. For instance, fix $Y=1_a 1_b 1_c 1_d$ with a generator $1_n$ of $\mathbb{Z}_n$. May $f$ not be a monomorphism?

Answer 1

The first exercise in Serre’s book Trees spells this out (I’ve corrected a missing “normal”):

Let $f_1\colon A \to G_1$ and $f_2\colon A \to G_2$ be two homomorphisms and let $G$ be their pushout. We define subgroups $A^n$, $G^n_1$ and $G^n_2$ of $A$, $G_1$ and $G_2$ recursively by the following conditions:

• $A^1 = \{1\}, \qquad G^1_1=\{1\}, \qquad G^1_2 = \{1\}$

• $A^n = $ normal subgroup of $A$ generated by $f^{-1}_1(G^{n-1}_1)$ and $f^{-1}_2(G^{n-1}_2)$

• $G^n_i = $ normal subgroup of $G_i$ generated by $f_i(A^n)$.

Let $A^\infty$, $G^\infty_i$ be the unions of the $A^n$, $G^n_i$, respectively. Show that $f_i$ defines an injection $A/A^\infty \to G_i/G^\infty_i$ and that $G$ may be identified with the amalgam of $G_1/G^\infty_1$ and $G_2/G^\infty_2$ along $A/A^\infty$.

It follows (using the results of no. 1.2 in Trees) that the kernel of $A\to G$ is $A^\infty$ and that the kernel of $G_i \to G$ is $G_i^\infty$.