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 → G a simple homomorphism, that is, the only subgroup K of H, normal in G and f-invariant (Kf ≤ K) is trivial. The group G then has two chains of subgroups: G0 = G, H0 = H, Gk = (Hk−1)f, Hk = H ∩ Gk for (k ≥ 1). We define a family of self-similar polycyclic groups, denoted SSP, where each subgroup Gk is self-similar with respect to the triple (Gk, Hk, f) for all k. By definition, a group G belongs to this SSP family provided f : H → G is a monomorphism, Hk and Gk+1 are normal subgroups of index p in Gk (p a prime or infinite) and Gk = HkGk+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 {a1, . . . , an} which is self-similar under the action of f : a1 → a2 → . . . → an, 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.