Federated Learning (FL) faces significant challenges related to communication efficiency and performance reduction when scaling to many clients. To address these issues, we explore the potential of using low-rank updates and provide the first theoretical study of rank properties in FL. Our theoretical analysis shows that a client's loss exhibits a higher-rank structure (i.e., gradients span higher-rank subspaces of the Hessian) compared to the server's loss, and that low-rank approximations of the clients' gradients have greater similarity. Based on this insight, we hypothesize that constraining client-side optimization to a low-rank subspace could provide an implicit regularization effect while reducing communication costs. Consequently, we propose FedLoRU, a general low-rank update framework for FL. Our framework enforces low-rank client-side updates and accumulates these updates to form a higher-rank model. We are able to establish convergence of the algorithm; the convergence rate matches FedAvg. Additionally, variants of FedLoRU can adapt to environments with statistical and model heterogeneity by employing multiple or hierarchical low-rank updates. Experimental results demonstrate that FedLoRU performs comparably to full-rank algorithms and exhibits robustness to heterogeneous and large numbers of clients.