Algebra Seminar talk
Reconstruction from Automorphism Groups - Rubin's approach
Given a relational structure, how much does its automorphism group know about the structure? In other words, if two structures have automorphism groups which are isomorphic (as abstract groups), how much are the two structures resembling each other? In general, the automorphism group contains very little information, but under suitable hypotheses, some remarkable results are known. Using his notion of so-called forall-exists-interpretations of structures, M.Rubin proved a particularly strong theorem: He gave conditions for the two structures from above to be "interdefinable", i.e. if viewed over a canonical expansion of the signatures, the structures are isomorphic. Examples for this kind of reconstruction include the rationals as well as the random graph. In my talk, I will give an overview of some conditions which are necessary for this kind of reconstruction to happen, define Rubin's notion of interpretation and motivate his theorem as well as provide the proof strategy. If time permits, I will also describe a possible method to show that a structure is within the realm of Rubin's theorem.