Maximal \(\ell\)-Frattini quotients of \(\ell\)-Poincaré duality groups of dimension 2. (Q2487031)

Let \(l\) be a prime number. An \(l\)-Frattini extension of a profinite group \(A\) is a surjective morphism \(\pi\colon B\to A\) of profinite groups whose kernel is contained in the pro-\(l\) Sylow subgroup of \(\text{Frat\,}B\). The author studies maximal \(l\)-Frattini extensions one can associate to every surjective morphism \(\varphi\colon\widehat G\to A\) of profinite groups. A pair \((\pi,\beta)\) of epimorphisms \(\pi\colon B\to A\), \(\beta\colon\widehat G\to B\) is an \(l\)-Frattini quotient of \(\varphi\) if \(\varphi=\pi\circ\beta\) and \(\pi\) is an \(l\)-Frattini extension. The \(l\)-Frattini quotient \((\pi,\beta)\) of \(\varphi\) is called maximal if every non-trivial minimal finite \(l\)-embedding problem for \(\beta\) has no weak solution (i.e. \(\pi\) is a maximal \(l\)-Frattini extension \(\varphi\) can factor through). If \(A\) is finite and \(\widehat G\) is a profinite group of cohomological dimension 1, then every maximal \(l\)-Frattini quotient of \(\varphi\) coincides with the universal \(l\)-Frattini extension; in contrast when \(G\) has cohomological dimension 2, there is no generic behaviour of maximal \(l\)-Frattini quotients. In this paper the case is studied when \(\widehat G\) is a weakly-orientable \(l\)-Poincaré duality group of dimension 2 and \(A\) is a finite group whose order is divisible by \(l\). This analysis can be applied to the study of modular towers. It is shown that the existence of finite maximal \(l\)-Frattini quotients is controlled by an integer \(r_l(A)\) that can be computed from the knowledge of the irreducible \(\overline\mathbb{F}_l[A]\)-modules. Moreover, the author studies properties of \(\varphi\) which imply that for every maximal \(l\)-Frattini quotient \((\pi,\beta)\), the group \(B\) itself is a weakly-orientable \(l\)-Poincaré duality group of dimension 2.