A group theoretic analogue of the Parshin-Arakelov rigidity theorem (Q1337763)

Let \({\mathfrak D}\) denote the class of fundamental groups of closed punctured (in finitely many points) topological surfaces. Let \(G\) be a group. The author calls a \({\mathfrak D}^2\) structure on \(G\) a normal \({\mathfrak D}\)-subgroup \(H\) such that \(G/H\) is also a \({\mathfrak D}\)-group. He conjectures that any group possesses at most finitely many \({\mathfrak D}^2\) structures. This conjecture is proven under some additional restrictions. It is also shown that for a \({\mathfrak D}^2\) structure \((G,H)\) the stabilizer \(\text{Stab}_G (H)\) is a subgroup of finite index in \(\Aut (G)\).