In the paper we deploy two related theoretical and methodological approaches to see how they complement each other in the analysis of oral communication in mathematics group work. The first approach is an authority framework that builds on positioning theory and the second is politeness theory, with a focus on the sociological notion of face. Authority and face both describe the relationship among individuals in interaction. We analyse two episodes of students interacting in group work, one instance with teacher mediation and one without. We use these to study how students' and teachers' face-needs are interwoven with authority structures. We ask how politeness theory and the authority framework complement each other, and how the associated conceptual tools illuminate each other when used together. Our combined analyses underline the importance of the authority of an imagined expert for the maintenance of smooth cooperation. At the same time, while the authority framework describes the acknowledgment of choices as an open dialogical space, politeness theory describes the awareness of choices as a potential threat. Both frameworks envision an alignment as a possible outcome of interactions that seem to contain conflicting face-needs.