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 M 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). © 2007 London Mathematical Society.