Related papers: Reduction principles for proper actions

We give a criterion for the rigidity of actions on homogeneous spaces. Let $G$ be a real Lie group, $\Lambda$ a lattice in $G$, and $\Gamma$ a subgroup of the affine group Aff$(G)$ stabilizing $\Lambda$. Then the action of $\Gamma$ on…

Consider a compact prequantizable symplectic manifold M on which a compact Lie group G acts in a Hamiltonian fashion. The ``quantization commutes with reduction'' theorem asserts that the G-invariant part of the equivariant index of M is…

In his seminal work \cite{pal:61}, R. Palais extended a substantial part of the theory of compact transformation groups to the case of proper actions of locally compact groups. Here we extend to proper actions some other important results…

We consider the pseudo-Riemannian Lichnerowicz conjecture in the homogeneous setting. In particular, we show that any compact connected pseudo-Riemannian manifold $M$ on which a semisimple group $G$ acts conformally, essentially and…

We prove a Lichnerowicz type vanishing theorem for non-compact spin manifolds admiting proper cocompact actions. This extends a previous result of Ziran Liu who proves it for the case where the acting group is unimodular.

We consider actions of real Lie subgroups G of complex reductive Lie groups on Kaehlerian spaces. Our main result is the openness of the set of semistable points with respect to a momentum map and the action of G.

For a proper, cocompact action by a locally compact group of the form $H \times G$, with $H$ compact, we define an $H \times G$-equivariant index of $H$-transversally elliptic operators, which takes values in $KK_*(C^*H, C^*G)$. This…

Let P be a semigroup that admits an embedding into a group G. Assume that the embedding satisfies a certain Toeplitz condition and that the Baum-Connes conjecture holds for G. We prove a formula describing the K- theory of the reduced…

For a proper action by a locally compact group $G$ on a manifold $M$ with a $G$-equivariant Spin-structure, we obtain obstructions to the existence of complete $G$-invariant Riemannian metrics with uniformly positive scalar curvature. We…

We generalize various symplectic reduction techniques to the context of the optimal momentum map. Our approach allows the construction of symplectic point and orbit reduced spaces purely within the Poisson category under hypotheses that do…

We prove that if $G$ is a noncompact connected real reductive linear Lie group, then any discrete subgroup of $G$ acting properly discontinuously and cocompactly on some homogeneous space $G/H$ of $G$ is quasi-isometrically embedded and…

Let G be a real reductive Lie group and H a closed reductive subgroup of G. We investigate the deformation of "standard" compact quotients of G/H, i.e., of quotients of G/H by discrete subgroups Gamma of G that are uniform lattices in a…

We prove a structure result on proper extensions of two-sided restriction semigroups in terms of partial actions, generalizing respective results for monoids and for inverse semigroups and upgrading the latter. We introduce and study…

We prove that an isometric action of a Lie group on a Riemannian manifold admits a resolution preserving the transverse geometry if and only if the action is infinitesimally polar. We provide applications concerning topological simplicity…

A Theorem due to Guillemin and Sternberg about geometric quantization of Hamiltonian actions of compact Lie groups $G$ on compact Kaehler manifolds says that the dimension of the $G$-invariant subspace is equal to the Riemann-Roch number of…

The main purpose of these lecture notes is to provide a concise introduction to Lie groups, Lie algebras, and isometric and adjoint actions, aiming mostly at advanced undergraduate and graduate students. In addition, the connection between…

We prove pointwise and maximal ergodic theorems for probability measure preserving (p.m.p.) actions of any countable group, provided it admits an essentially free, weakly mixing amenable action of stable type $III_1$. We show that this…

We consider a connected symplectic manifold $M$ acted on by a connected Lie group $G$ in a Hamiltonian fashion. If $G$ is compact, we prove give an Equivalence Theorem for the symplectic manifolds whose squared moment map $\parallel \mu…

If $G$ is a compact group acting continuously on a compact metric space $(X, m)$, we prove two results that generalize Dirichlet's classical theorem on Diophantine approximation. If $G$ is a noncommutative compact group of isometries, we…

We discuss properties of orbits of (semi)group actions on locally compact groups G. In particular, we show that if a compactly generated locally compact abelian group acts distally on G then the closure of each of its orbits is a minimal…