Analytic stacks and hyperbolicity

In this article we give two notions of hyperbolicity for groupoids on the analytic site of complex spaces, which we call Kobayashi and Brody hyperbolicity. In the special case the groupoid is a complex analytic space, these notions of hyperbolicity give the classical ones due to Kobayashi and Brody. We prove that such notions are equivalent if the groupoid is a compact Deligne–Mumford analytic stack (in analogy with the Brody theorem). Moreover, under the same assumptions, such notions of hyperbolicity are completely detected by the coarse moduli space of the stack. We finally show that stack hyperbolicity, as we defined it, is expected to impose a peculiar behavior to the stack itself, much like hyperbolicity for complex spaces. For instance, a stronger notion of it (hyperbolicity of the coarse moduli space) implies a “strong asymmetry” on the stack in the compact case, namely that its automorphism 2-group has only finitely many isomorphism classes.