On a Class of Self-Similar Polycyclic Groups

TL;DR

The paper develops a structural theory of self-similar polycyclic groups (SSP) by introducing chains $G_k$, $H_k$ derived from a simple triple $(G,H,f)$ with $f$ injective and $[G:H]=p$, and requiring $G_k=H_kG_{k+1}$ with $G_k/H_k$ cyclic of order $p$ or infinite. It proves that finite SSP groups are exactly finite $p$-groups with a simple monomorphism, and shows that SSP groups have a canonical self-similar generating set of Hirsch length $n$, yielding a nilpotent metabelian structure with center rank at least $n/3$. The authors classify metabelian SSPs via $G(n,c)$-type parameterization, where $n=3q(n)+r(n)$ and $c=c(n,j)=2q(n)+r(n)+j$, and prove that non-abelian SSP groups of this type have nilpotency controlled by $n$ and $c$, with a detailed center decomposition. They further analyze the nilpotency class, provide matrix representations for class-2 nilpotent SSPs, and study liftings to higher $n$-types, obtaining a complete picture in the $p=2$ case where the lifted groups have nilpotency class at most $2$, while describing lift obstructions for odd primes. These results establish a robust framework for constructing and classifying self-similar, metabelian, polycyclic groups and reveal a delicate modulo-3 arithmetic structure governing their center and nilpotency properties.

Abstract

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.

Theorems & Definitions (11)

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