Reconstruction of classical geometries from their automorphism group

Barbina, S. (2007). Reconstruction of classical geometries from their automorphism group. Journal of the London Mathematical Society, 75(2) pp. 298–316.

DOI: https://doi.org/10.1112/jlms/jdl016

Abstract

Let V be a countably infinite-dimensional vector space over a finite field F. Then V is omega-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 omega-categorical structure such that Aut(M) and Aut(N) are isomorphic 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).

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