This paper is concerned with the study of Besov-type decomposition spaces, which are scales of spaces associated to suitably defined coverings of the euclidean space R-d, or suitable open subsets thereof. A fundamental problem in this domain, that is currently not fully understood, is deciding when two different coverings give rise to the same scale of decomposition spaces. In this paper, we establish a coarse geometric approach to this problem, and show how it specializes for the case of wavelet coorbit spaces associated to a particular class of matrix groups H H-2 is a quasi-isometry with respect to suitably defined word metrics. We then proceed to apply this criterion to a large class of dilation groups called shearlet dilation groups, where this quasi-isometry condition can be characterized algebraically. We close with the discussion of selected examples. (C) 2022 Elsevier Inc. All rights reserved.
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