Over the past few years, there has been a substantial effort towards automated detection of fake news on social media platforms. Existing research has modeled the structure, style, content, and patterns in dissemination of online posts, as well as the demographic traits of users who interact with them. However, no attention has been directed towards modeling the properties of online communities that interact with the posts. In this work, we propose a novel social context-aware fake news detection framework, SAFER, based on graph neural networks (GNNs). The proposed framework aggregates information with respect to: 1) the nature of the content disseminated, 2) content-sharing behavior of users, and 3) the social network of those users. We furthermore perform a systematic comparison of several GNN models for this task and introduce novel methods based on relational and hyperbolic GNNs, which have not been previously used for user or community modeling within NLP. We empirically demonstrate that our framework yields significant improvements over existing text-based techniques and achieves state-of-the-art results on fake news datasets from two different domains.