Abstract

Let \( \mathcal{C} \) be a class of groups. We say that \textbf{profinite completions detect free products} in \( \mathcal{C} \) if every finitely generated residually finite group \( G \in \mathcal{C} \), whose profinite completion \( \widehat{G} \) decomposes as \( \widehat{G} = L \amalg K \) with non-trivial \( L \) and \( K \), also decomposes as a free product \( G = H U \) with non-trivial subgroups \( H \) and \( U \). We also say that \textbf{profinite completions detect free factors} in \( \mathcal{C} \) if, for every finitely generated residually finite group \( G \in \mathcal{C} \) with a finitely generated subgroup \( H \), such that the profinite completion \( \widehat{G} \) decomposes as \( \widehat{G} = \overline{H} \amalg K \), there exists a subgroup \( U \leq G \) such that \( G = H U \). I will discuss this property in the class of fundamental groups of graphs of virtually free groups with virtually cyclic edge groups. The talk is based on work in progress with Henrique Souza and Pavel Zalesski.