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A group theoretic analogue of the Parshin-Arakelov rigidity theorem. (English) Zbl 0829.57001

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 \(\operatorname{Aut} (G)\).

MSC:

57M05 Fundamental group, presentations, free differential calculus
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
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References:

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