@article{Ionescu_Kumjian_Renault_Sims_Williams_2021, title={Pushouts of extensions of groupoids by bundles of abelian groups}, volume={52}, url={https://nzjmath.org/index.php/NZJMATH/article/view/136}, DOI={10.53733/136}, abstractNote={<p>We analyse extensions $\Sigma$ of groupoids G by bundles A of abelian groups. We describe a pushout construction for such extensions, and use it to describe the extension group of a given groupoid G by a given bundle A. There is a natural action of Sigma on the dual of A, yielding a corresponding transformation groupoid. The pushout of this transformation groupoid by the natural map from the fibre product of A with its dual to the Cartesian product of the dual with the circle is a twist over the transformation groupoid resulting from the action of G on the dual of A. We prove that the full C*-algebra of this twist is isomorphic to the full C*-algebra of $\Sigma$, and that this isomorphism descends to an isomorphism of reduced algebras. We give a number of examples and applications.</p>}, journal={New Zealand Journal of Mathematics}, author={Ionescu, Marius and Kumjian, Alex and Renault, Jean N. and Sims, Aidan and Williams, Dana P.}, year={2021}, month={Sep.}, pages={561–581} }