The goal of this talk is to present an algorithm which takes a compact square complex belonging to a special class as input and decides whether its fundamental group splits as a free product. The special class is built by attaching tubes to finite graphs in such a way that they satisfy a nonpositive curvature condition. This construction gives rise to a rich class of complexes, including, but not limited to, closed surfaces of positive genus. The algorithm can be used to deduce the celebrated Stallings theorem for this special class, as also the well known Grushko decomposition theorem.

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Last updated: 29 Nov 2023