Rigid geometric structures, isometric actions, and algebraic quotients

TL;DR

This paper extends Gromov's theorems on isometric actions of Lie groups with rigid structures, using algebraic quotient techniques to reveal new structural and fixed point properties, especially for split solvable groups.

Contribution

It generalizes Gromov's centralizer and representation theorems to broader classes of Lie groups and provides new fixed point and structural results for isometric actions on manifolds.

Findings

Generalized Gromov's theorems to split solvable groups

Proved structural properties of isometry groups for certain manifolds

Established fixed point theorems for split solvable group actions

Abstract

By using a Borel density theorem for algebraic quotients, we prove a theorem concerning isometric actions of a Lie group $G$ on a smooth or analytic manifold $M$ with a rigid $\mathrm{A}$-structure $\sigma$. It generalizes Gromov's centralizer and representation theorems to the case where $R(G)$ is split solvable and $G/R(G)$ has no compact factors, strengthens a special case of Gromov's open dense orbit theorem, and implies that for smooth $M$ and simple $G$, if Gromov's representation theorem does not hold, then the local Killing fields on $\widetilde{M}$ are highly non-extendable. As applications of the generalized centralizer and representation theorems, we prove (1) a structural property of $\mathrm{Iso}(M)$ for simply connected compact analytic $M$ with unimodular $\sigma$, (2) three results illustrating the phenomena that if $G$ is split solvable and large then $\pi_1(M)$ is also large, and (3) two fixed point theorems for split solvable $G$ and compact analytic $M$ with non-unimodular $\sigma$.