Let $G$ be a group. Recall that a group $H$ is called a retract of $G$ if there exist homomorphisms $g:G\longrightarrow{H}$ and $f:H\longrightarrow G$ so that $g\circ f=id_H$. The homomorphism $g$ is called a retraction.

My question is that:

Is there an infinite sequence $\{ H_i \}_{i\in \mathbb{N}}$ of groups so that the following conditions hold?

1- $H_1$ is a finitely presented group.

2- $H_{i+1}$ is proper retract of $H_i$ with the retraction $g_{i}:H_i \longrightarrow H_{i+1}$, for all $i\geq 1$ (equivalently, $H_i$ admits a semidirect product decomposition $L_i\ltimes N_i$ with $N_i\neq 1$ and $L_i\simeq H_{i+1}$);

Thanks in advance.

8more comments