  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.
