Título: On a Class of Self-Similar Polycyclic Groups

Palestrante: Emerson de Melo (UnB)

Resumo: 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. Recently, we introduced a family of self-similar polycyclic groups called SSP. In particular, if G is a finite SSP-group, then G is a finite p-group with [G : H] = p and f injective. In this talk, we present this family and its main properties. Surprisingly, if the Hirsch length of an SSP-group is n, then the arithmetic of n modulo 3 has a strong impact on the structure. This fact allows us to prove that an SSP-group is nilpotent metabelian whose center is free p-abelian (p prime or infinite) of rank at least n/3. Furthermore, when G is an SSP 2-group, we prove that G has nilpotency class at most 2, and classify all such groups.