Martin Pergler - Connection preserving actions and observable and epimorphic subgroups

Let G be a real algebraic group. We consider which (algebraic) subgroups H arise as point stabilizers in affine connection preserving G-actions on manifolds, and for which H are any H-fixed points of such actions necessarily G-fixed points.

We prove that under certain hypotheses (conjecturally always) these are the same H with analogous properties concerning invariant vectors in (finite-dimensional) linear representations, called observable and epimorphic subgroups. Techniques involve contrasting local dynamics and linearization of the action near fixed points, together with the structure theory of epimorphic subgroups. The result forms a part of Zimmer's program of studying representations of Lie groups into the automorphism groups of geometric structures on manifolds.

We also extend results of Bien and Borel to classify all epimorphic subgroups of SL_n normalized by a maximal torus, in terms of graphs on n vertices and a certain subset of the Tits boundary.