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**1 - 1**of**1**### THE AFFINE AND PROJECTIVE GROUPS ARE MAXIMAL

"... Abstract. We show that the groups AGLn(Q) and PGLn(Q), seen as closed subgroups of S∞, are maximal-closed. This work is part of a long line of research studying reducts of first order structures, or dually extensions of closed-subgroups of S∞. Most of the research so far has focused on the ω-cateori ..."

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Abstract. We show that the groups AGLn(Q) and PGLn(Q), seen as closed subgroups of S∞, are maximal-closed. This work is part of a long line of research studying reducts of first order structures, or dually extensions of closed-subgroups of S∞. Most of the research so far has focused on the ω-cateorical case where the two points of view (group or structure) are equivalent: a countable ω-categorical structure is determined up to bi-interpretability by its topological automorphism group G < S ∞ and its reducts are in one-to-one correspondence with closed subgroups of S ∞ containing G [AZ86]. The full classification of reducts is known for a number of ω-categorical structures ([Cam76, Tho91, Tho96, BP11]). Nevertheless, the main question asked by Thomas over 20 years ago remains unresolved: we do not know if every homogeneous structure on a finite relational language has only finitely many reducts. In an other direction, Junker and Ziegler [JZ08] asked for a converse to that question: if a structure is not ω-categorical, does it necessarily have infinitely many reducts. One has to beware when considering non ω-categorical structures that there are now two natural notions of reducts. Let M be a countable structure in a language L and let GM < S ∞ be its automorphism group. A