  Hyperbolic models are remarkably good at reproducing the scale-free, highly
clustered and small-world properties of networks representing real complex
systems in a very simple framework. Here we show that for the
popularity-similarity optimization model from this family, the generated
networks become also extremely modular in the thermodynamic limit, in spite of
lacking any explicit community formation mechanism in the model definition.
According to our analytical results supported by numerical simulations, when
the system size is increased, the modularity approaches one surprisingly fast.
