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Universal covering spaces and fundamental groups in algebraic geometry as schemes

Ravi Vakil;  Kirsten Wickelgren — 2011

Journal de Théorie des Nombres de Bordeaux

In topology, the notions of the fundamental group and the universal cover are closely intertwined. By importing usual notions from topology into the algebraic and arithmetic setting, we construct a fundamental group family from a universal cover, both of which are schemes. A geometric fiber of the fundamental group family (as a topological group) is canonically the étale fundamental group. The constructions apply to all connected quasicompact quasiseparated schemes. With different methods and hypotheses,...

On weakly mixing and doubly ergodic nonsingular actions

Sarah Iams;  Brian Katz;  Cesar E. Silva;  Brian Street;  Kirsten Wickelgren — 2005

Colloquium Mathematicae

We study weak mixing and double ergodicity for nonsingular actions of locally compact Polish abelian groups. We show that if T is a nonsingular action of G, then T is weakly mixing if and only if for all cocompact subgroups A of G the action of T restricted to A is weakly mixing. We show that a doubly ergodic nonsingular action is weakly mixing and construct an infinite measure-preserving flow that is weakly mixing but not doubly ergodic. We also construct an infinite measure-preserving flow whose...

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