Robust Hilbert space fragmentation in group-valued loop models

We introduce a large class of models exhibiting robust ergodicity breaking in quantum dynamics. Our work is inspired by recent discussions of "topologically robust Hilbert space fragmentation," but massively generalizes in two directions: firstly from states describable as "loop-soups" to a broader class of states reminiscent of string-nets and sponges, and secondly from models restricted to square or cubic lattices, to models defined on arbitrary lattices (and even arbitrary graphs without translation invariance). Our constructions leverage a recently proposed group-theory framework [PRX 14, 021034 (2024)], and identify a host of new phenomena arising from the interplay of "group-model dynamics" and lattice structure. We make crisp connections to gauge theories, and our construction generalizes Kitaev's quantum double to infinite groups.