Introduced by Breuillard and Sert, the joint spectrum of a finite set of matrices records the asymptotics of all possible tuples of singular values of products of matrices in the set. Under reasonable assumptions, this set is compact, has non-empty interior, and can be recovered from the set of Lyapunov vectors for random walks supported on the set. I will discuss joint work with Stephen Cantrell and Cagri Sert, in which we define the joint translation spectrum, an analogous object for word hyperbolic groups. The input now is a tuple of geometric actions of the group on geodesic metric spaces. In this setting, we prove a version of Breuillard-Sert's theorem: the joint translation spectrum is recovered from the set of drift vectors for admissible random walks on the group.