# Decomposition complexity growth of finitely generated groups

## TL;DR

This paper introduces decomposition complexity growth, a new invariant that generalizes existing concepts like finite decomposition complexity and dimension growth, and shows its implications for property A in finitely generated groups.

## Contribution

The paper defines decomposition complexity growth as a quasi-isometry invariant and demonstrates its significance in understanding property A and group constructions.

## Key findings

- Subexponential decomposition complexity growth implies property A.
- Decomposition complexity growth is preserved under certain group and metric constructions.
- The notion generalizes finite decomposition complexity and dimension growth.

## Abstract

Finite decomposition complexity and asymptotic dimension growth are two generalizations of M. Gromov's asymptotic dimension which can be used to prove property A for large classes of finitely generated groups of infinite asymptotic dimension. In this paper, we introduce the notion of decomposition complexity growth, which is a quasi-isometry invariant generalizing both finite decomposition complexity and dimension growth. We show that subexponential decomposition complexity growth implies property A, and is preserved by certain group and metric constructions.