Linearization of proper group actions

Linearization of Regular Proper Groupoids

Let G be a Lie groupoid over M such that the target-source map from G to M ×M is proper. We show that, if O is an orbit of finite type (i.e. which admits a proper function with finitely many critical points), then the restriction G|U of G to some neighborhood U of O in M is isomorphic to a similar restriction of the action groupoid for the linear action of the transitive groupoid G|O on the nor...

Proper actions and proper invariant metrics

We show that if a locally compact group G acts properly on a locally compact σ-compact space X, then there is a family of G-invariant proper continuous finite-valued pseudometrics which induces the topology of X. If X is, furthermore, metrizable, then G acts properly on X if and only if there exists a G-invariant proper compatible metric on X.

Singular Reduction for Proper Actions

Geometric Quantization for Proper Actions

We first introduce an invariant index for G-equivariant elliptic differential operators on a locally compact manifold M admitting a proper cocompact action of a locally compact group G. It generalizes the Kawasaki index for orbifolds to the case of proper cocompact actions. Our invariant index is used to show that an analog of the Guillemin-Sternberg geometric quantization conjecture holds if M...

Proper Actions of Groupoids on C-algebras

This thesis contains some results concerning groupoid dynamical systems and crossed products. We introduce the notion of a proper groupoid dynamical system and of its generalized fixed point algebra. We show that our notion of proper groupoid dynamical system extends both the notion of proper actions of groups on topological spaces and the notion of the proper group dynamical systems introduced...