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The class of generalized shearlet dilation groups has recently been developedto allow the unified treatment of various shearlet groups and associatedshearlet transforms that had previously been studied on a case-by-case basis.We consider several aspects of these groups: First, their systematicconstruction from associative algebras, secondly, their suitability for thecharacterization of wavefront sets, and finally, the question of constructingembeddings into the symplectic group in a way that intertwines thequasi-regular representation with the metaplectic one. For all questions, it ispossible to treat the full class of generalized shearlet groups in acomprehensive and unified way, thus generalizing known results to an infinityof new cases. Our presentation emphasizes the interplay between the algebraicstructure underlying the construction of the shearlet dilation groups, thegeometric properties of the dual action, and the analytic properties of theassociated shearlet transforms.

Recent Progress in Shearlet Theory: Systematic Construction of Shearlet Dilation Groups, Characterization of Wavefront Sets, and New Embeddings