arXiv subjects
Publications and source records attributed to Scot Adams.
We provide counterexamples to P. Olver's freeness conjecture for $C^\infty$ actions. In fact, we show that a counterexample exists for any connected real Lie group with noncompact center, as well as for the additive group of the integers.
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.
Let a real Lie group $G$ act on a $C^\infty$ real manifold $M$. Assume that the action is $C^\infty$ and that every nontrivial element of $G$ has a nowhere dense fixpoint set in $M$. We show that, in~some higher order frame bundle $F$ of $M$, there exists a $G$-invariant meager subset $Z$ of $F$ such that the $G$-action on $F\backslash Z$ is free. A similar result holds for submanifold jet bundles.
Let $G$~be a real Lie group and let $G^\circ$ be the identity component of~$G$. Let $G$~act on a $C^\infty$ real manifold~$M$. Assume the action is $C^\infty$. Assume that the fixpoint set of any nontrivial element of~$G^\circ$ has empty interior in~$M$. Let $n:=\dim G$. Assume $n\ge1$. Let $F$ be the frame bundle of~$M$ of order $n-1$. We prove: there exists a $G$-invariant dense open subset~$Q$ of~$F$ such that the $G$-action on $Q$ has discrete stabilizers.
We prove that every real algebraic action on a smooth real algebraic variety has a prolongation with a "moving frame".
We provide a counterexample to P.~Olver's freeness conjecture for $C^ω$ transformations.
We prove that if a unitary representation of a connected Lie group has the property that no nonozero vector is fixed by any nontrival normal connected subgroup, then the matrix coefficients decay to zero at Adjoint-infinity.