Reconstruction of classical geometries from their automorphism group

Let V be a countably infinite‐dimensional vector space over a finite field F . Then V is ω‐categorical, and so are the projective space PG ( V ) and the projective symplectic, unitary and orthogonal spaces on V . Using a reconstruction method developed by Rubin, we prove the following result: let ℳ be one of the above spaces, and let be an ω‐categorical structure such that Aut (ℳ) ≅ Aut () as abstract groups. Then ℳ and are bi‐interpretable. We also give a reconstruction result for the affine group AGL ( V ) acting on V by proving that V as an affine space is interpretable in AGL ( V ).