Decompositions of fundamental groups of closed surfaces into free constructions (Q1855281)

In this paper it is proven that for a closed surface \(T\) any decomposition of \(\pi_1(T,x)\) into an amalgamated product (or, more generally, into the fundamental group of a finite graph of groups) with f.g. edge group(s) is almost geometric. This means that there is a subgroup \(H\) of fintie index in \(\pi_1(T,x)\) such that the induced decomposition of \(H\) is geometric in the corresponding covering of \(T\). In addition, the author investigates to what extent the edge groups determine the vertex groups.