The Root-Class Residuality of the Fundamental Groups of Graphs of Groups

Consider a class $ \mathcal{C} $ of groups containing at least one nontrivial group and closed under subgroups, extensions, and Cartesian products of the form $ \prod\nolimits_{y\in Y}X_{y} $ , where $ X,Y\in{\mathcal{C}} $ and $ X_{y} $ is an isomorphic copy of $ X $ for each $ y\in Y $ . Given an arbitrary graph of groups, we obtain necessary and sufficient conditions for its fundamental group $ G $ to be a residually $ \mathcal{C} $ -group thus generalizing Shirvani’s conditions for the residual finiteness of $ G $ .