# A method for determining Cartan geometries from the local behavior of automorphisms

## TL;DR

This paper presents a new method to construct Cartan geometries from the local automorphism behavior, enabling comparison and extension of automorphisms in various geometric structures.

## Contribution

It introduces the 'sprawl' construction for Cartan geometries that captures local automorphism behavior and provides a universal property for comparing different geometries.

## Key findings

- Constructed Cartan geometries from local automorphism data
- Extended local automorphisms to global automorphisms in specific geometries
- Provided examples in projective and parabolic geometries

## Abstract

We introduce a construction for a Cartan geometry that captures the local behavior of a given geometric automorphism near a distinguished element. The result of this construction, which we call the sprawl generated by the automorphism, is uniquely characterized by a kind of universal property that allows us to compare different Cartan geometries that admit automorphisms with equivalent local behavior near a distinguished element. As example applications, we describe how to construct non-flat real projective structures admitting nontrivial automorphisms with higher-order fixed points and extend some known local automorphisms with higher-order fixed points on non-flat parabolic geometries to global automorphisms.