This work is motivated by two problems: 1) The approach of manifolds and spaces by triangulations. 2) The complexity growth in sequences of polyhedra. Considering both problems as related, new criteria and methods for approximating smooth manifolds are deduced. When the sequences of polyhedra are obtained by the action of a discrete group or semigroup, further control is given by geometric, topologic and complexity observables. We give a set of relevant examples to illustrate the results, both in infinite and finite dimensions.
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