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We consider the coorbit theory associated to general continuous wavelettransforms arising from a square-integrable, irreducible quasi-regularrepresentation of a semidirect product group $G = \mathbb{R}^d \rtimes H$. Theexistence of coorbit spaces for this very general setting has been recentlyestablished, together with concrete vanishing moment criteria for analyzingvectors and atoms that can be used in the coorbit scheme. These criteria dependon fairly technical assumptions on the dual action of the dilation group, andit is one of the chief purposes of this paper to considerably simplify theseassumptions.  We then proceed to verify the assumptions for large classes of dilationgroups, in particular for all abelian dilation groups, as well as a classcalled {\em generalized shearlet dilation groups}, containing and extending allknown examples of shearlet dilation groups employed in dimensions two andhigher. We explain how these groups can be systematically constructed fromcertain commutative associative algebras of the same dimension, and give a fulllist, up to conjugacy, of shearing groups in dimensions three and four. In thelatter case, three previously unknown groups arise.  As a result the existence of Banach frames consisting of compactly supportedwavelets, with simultaneous convergence in a whole range of coorbit spaces, isestablished for all groups involved.

Simplified vanishing moment criteria for wavelets over general dilation groups, with applications to abelian and shearlet dilation groups