Étale groupoids and Steinberg Algebras : a concise introduction

In the last couple of years, étale groupoids have become a focal point in several areas of mathematics. The convolution algebras arising from étale groupoids, considered both in analytical setting [50] and algebraic setting [23, 54], include many deep and important examples such as Cuntz algebras [27] and Leavitt algebras [40] and allowsystematic treatment of them. Partial actions and partial symmetries can also be realised as étale groupoids (via inverse semigroups), allowing us to relate convolution algebras to partial crossed products [28, 30]. Realising that the invariants long studied in topological dynamics can be modelled on étale groupoids (such as homology, full groups and orbit equivalence [41]) and that these are directly related to invariants long studied in analysis and algebra (such as K-theory) allows interaction between areas; we can use techniques developed in algebra in analysis and vice versa. The étale groupoid is the Rosetta stone.