The universal Euler characteristic and Burnside group for definable groupoids

Abstract

We introduce the universal Euler characteristic of orbit space definable groupoids, a class of groupoids containing cocompact proper Lie groupoids as well as translation groupoids associated to proper definable group actions. We show that every additive and multiplicative invariant of orbit space definable groupoids with an additional local triviality hypothesis arises as a ring homomorphism applied to the universal Euler characteristic. This in particular includes the $\Gamma$-orbifold Euler characteristic introduced by the first and third authors when $\Gamma$ is a finitely presented group. For definable groupoids, where the object and arrow spaces as well as the structure maps are definable, we also introduce a Burnside group (which admits a partial multiplication), which generalizes the classical Burnside ring associated to compact Lie groups.