Rigid geometric structures, isometric actions, and algebraic quotients

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 $\sigma$.