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Finite decomposition complexity and asymptotic dimension growth are twogeneralizations of M. Gromov's asymptotic dimension which can be used to proveproperty A for large classes of finitely generated groups of infiniteasymptotic dimension. In this paper, we introduce the notion of decompositioncomplexity growth, which is a quasi-isometry invariant generalizing both finitedecomposition complexity and dimension growth. We show that subexponentialdecomposition complexity growth implies property A, and is preserved by certaingroup and metric constructions.

Decomposition complexity growth of finitely generated groups