An embedding theorem for automorphism groups of Cartan geometries

TL;DR

This paper establishes an embedding theorem linking automorphism groups of Cartan geometries to their model groups, leading to bounds on automorphism group properties and conditions for local homogeneity and volume-preserving actions.

Contribution

It introduces a new embedding theorem for automorphism groups of Cartan geometries, with implications for their structure and symmetry properties.

Findings

Bounds on rank and nilpotence degree of automorphism groups

Conditions for local homogeneity and completeness in parabolic geometries

A local freeness theorem for volume-preserving automorphism actions

Abstract

We prove a theorem relating the automorphism group of a Cartan geometry to the group on which the geometry is modeled: a component of the adjoint representation of the first embeds in the adjoint representation of the second. Consequences of the theorem include general bounds on the rank and nilpotence degree of an automorphism group; a result asserting local homogeneity and completeness of parabolic geometries admitting a maximal-rank group of automorphisms; and a local freeness theorem for actions additionally preserving a continuous volume form.