On a Class of Self-Similar Polycyclic Groups

A group $G$ is self-similar if it admits a triple $(G,H,f)$ where $H$ is a subgroup of $G$ and $f: H \to G$ a simple homomorphism, that is, the only subgroup $K$ of $H$, normal in $G$ and $f$-invariant ($K^f \leq K$) is trivial. The group $G$ then has two chains of subgroups: $$G_0 = G,\ H_0 = H,\ G_k = (H_{k-1})^f,\ H_k = H \cap G_{k}\ \text{for } (k \geq 1).$$ We define a family of self-similar polycyclic groups, denoted $SSP$, where each subgroup $G_k$ is self-similar with respect to the triple $(G_k , H_k, f)$ for all $k$. By definition, a group $G$ belongs to this $SSP$ family provided $f: H \rightarrow G$ is a monomorphism, $H_k$ and $G_{k+1}$ are normal subgroups of index $p$ in $G_k$ ($p$ a prime or infinite) and $G_k=H_kG_{k+1}$. When $G$ is a finite $p$-group in the class $SSP$, we show that the above conditions follow simply from $[G:H] = p$ and $f$ is a simple monomorphism. We show that if the Hirsch length of $G$ is $n$, then $G$ has a polycyclic generating set $\{a_1, \ldots, a_n\}$ which is self-similar under the action of $f: a_1 \rightarrow a_2 \rightarrow \ldots \rightarrow a_n$, and then $G$ is either a finite $p$-group or is torsion-free. Surprisingly, the arithmetic of $n$ modulo $3$ has a strong impact on the structure of $G$. This fact allows us to prove that $G$ is nilpotent metabelian whose center is free $p$-abelian ($p$ prime or infinite) of rank at least $n/3$. We classify those groups $G$ where $H$ has nilpotency class at most $2$. Furthermore, when $p=2$, we prove that $G$ is a finite $2$-group of nilpotency class at most $2$, and classify all such groups.