Discrete free Abelian central stabilizers in a higher order frame bundle

TL;DR

This paper investigates the structure of stabilizers in higher order frame bundles under smooth group actions, showing they are discrete, finitely-generated, free-Abelian, and central, with implications for the geometry of the action.

Contribution

It establishes that stabilizers in higher order frame bundles are discrete, free-Abelian, and central, extending understanding of group actions on manifolds.

Findings

Stabilizers in most frame bundles have no nontrivial compact subgroups.

For connected groups, a dense open subset has stabilizers that are discrete, finitely-generated, free-Abelian, and central.

Several geometric corollaries follow from these stabilizer properties.

Abstract

Let a real Lie group $G$ have a $C^\infty$ action on a real manifold $M$. Assume every nontrivial element of $G$ has nowhere dense fixpoint set in $M$. First, we show, in every frame bundle, except possibly the $0$th, that each stabilizer admits no nontrivial compact subgroups. Second, we show that, if $G$ is connected, then there is a dense open $G$-invariant subset of some higher order frame bundle of $M$ such that, for any point $x$ in that subset, the stabilizer in $G$ of $x$ is a discrete, finitely-generated, free-Abelian, central subgroup of $G$. We derive several corollaries of these two results.