Motivated by the search for new observables in nonperturbative quantum gravity, we consider Causal Dynamical Triangulations (CDT) in 2+1 dimensions with the spatial topology of a torus. This system is of particular interest, because one can study not only the global scale factor, but also global shape variables in the presence of arbitrary quantum fluctuations of the geometry. Our initial investigation focusses on the dynamics of the scale factor and uncovers a qualitatively new behaviour, which leads us to investigate a novel type of boundary conditions for the path integral. Comparing large-scale features of the emergent quantum geometry in numerical simulations with a classical minisuperspace formulation, we find partial agreement. By measuring the correlation matrix of volume fluctuations we succeed in reconstructing the effective action for the scale factor directly from the simulation data. Apart from setting the stage for the analysis of shape dynamics on the torus, the new set-up highlights the role of nontrivial boundaries and topology.