The Extension Property for Partial Automorphisms (EPPA) of Reducts of Relational Structures

Abstract

The Extension Property for Partial Automorphisms (EPPA), also called Hrusovski property, is a crucial concept in the realms of combinatorics, group theory, and model theory, linking the properties of structures and the classes of finitely generated substruc- tures that embed into them. The notion of EPPA, established by Hodges, Hodkinson, Lascar, and Shelah, has spurred significant advancements in understanding graph struc- tures and the automorphism groups associated with them. A milestone was achieved by Hrusovski, who demonstrated EPPA for the class of finite graphs. The research since has centered on categorizing more classes with EPPA, simplifying proof techniques, and understanding the broader implications of EPPA. This thesis contributes to this ongoing pursuit, specifically aiming to demonstrate EPPA for graph classes enriched by comple- mentary automorphisms. It includes an analysis of undirected and directed graphs with loops and extends the exploration to a class of general structures in a finite relational language. 1