Reconstruction of classical geometries from their automorphism group

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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 𝒩 be an ω-categorical structure such that Aut (ℳ) ≅ Aut (𝒩) as abstract groups. Then ℳ and 𝒩 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).