A method for determining Cartan geometries from the local behavior of automorphisms
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.
Keywords
Cite
@article{arxiv.2303.00561,
title = {A method for determining Cartan geometries from the local behavior of automorphisms},
author = {Jacob W. Erickson},
journal= {arXiv preprint arXiv:2303.00561},
year = {2025}
}
Comments
31 pages, 6 figures. Main results of original version were moved to arXiv:2406.05497. Revisions: Applications section has been redone to focus on non-flat parabolic examples