Gestures have been shown to play a key role in mathematical reasoning and to be an indicator that mathematical understanding is "embodied" -- inherently linked to action, perception, and the physical body. As learners collaborate and engage in mathematical discussions, they use discourse practices like explaining, refuting, and building on each other's reasoning. Here we examine how gestural embodied actions become distributed over multiple learners confronting mathematical tasks. We define "collaborative gestures" as gestural exchanges that take place as learners discuss and explore mathematical ideas, using their bodies in concert to accomplish a shared goal. We identify several ways in which learners' gestures can be used collaboratively and explore patterns in how collaborative gestures arise while proving geometric conjectures. Learners use collaborative gestures to extend mathematical ideas over multiple bodies as they explore, refine, and extend each other's mathematical reasoning. With this work, we seek to add to notions of important "talk moves" in mathematical discussions to also include a consideration of important "gesture moves". [This is the in press version of an article published in "Journal of Mathematical Behavior."]