# Topology of automorphism groups of parabolic geometries

## TL;DR

This paper proves that the automorphism group of any parabolic geometry can be given a Lie group structure where the $C^0$ and $C^{inite}$ topologies coincide, and it is closed within the homeomorphism group of the manifold.

## Contribution

It establishes the Lie group structure of automorphism groups of parabolic geometries and their topological properties, unifying their $C^0$ and $C^{inite}$ topologies.

## Key findings

- Automorphism group admits a Lie group structure.
- The $C^0$ and $C^{inite}$ topologies coincide.
- Automorphism group is closed in the homeomorphism group.

## Abstract

We prove for the automorphism group of an arbitrary parabolic geometry that the $C^0$ and $C^{\infty}$ topologies coincide, and the group admits the structure of a Lie group in this topology. We further show that this automorphism group is closed in the homeomorphism group of the underlying manifold.