Maximal ℓ-Frattini quotients of ℓ-Poincaré duality groups of dimension 2

Abstract.

Any morphism $\phi :\ifmmode\expandafter\hat\else\expandafter\^\fi{G} \to A$ of profinite groups has maximal ℓ-Frattini quotients

$$(\pi ,\beta ),\pi :B \to A,\;\beta :\ifmmode\expandafter\hat\else\expandafter\^\fi{G} \to B,\;{\text{i}}{\text{.e}}{\text{., }}\phi = \pi \circ \beta, \pi$$

is an ℓ-Frattini extension and β is a surjective morphism of profinite groups for which every minimal finite non-trivial ℓ-embedding problem is not weakly solvable. In this paper the case is studied where Ĝ Ĝ is a weakly-orientable ℓ-Poincaré duality group of dimension 2 and where A is a finite group whose order is divisible by ℓ. This analysis can be applied for the study of modular towers (Theorem A, Remark 1.2). It is shown that the existence of finite maximal ℓ-Frattini quotients is controlled by an integer r _ℓ(A) (Theorem B). In the final section we study properties of the morphism ϕ which imply that for every maximal ℓ-Frattini quotient (π, β), the profinite group B itself is a weakly-orientable ℓ-Poincaré duality group of dimension 2 (Theorem C).