Simplicity of some automorphism groups.
This article deals with the automorphism groups of homogeneous (relational) structures \(M\). These were in various special cases known to be simple; this is no general truth but as the authors point out in the introduction ``if \(\Aut(M)\) is not simple, then it has some \textit{obvious} proper non-trivial normal subgroups, explicable in terms of its actions''. The article provides for the first time a general result explaining the tendency to simplicity such groups have. It is shown (Theorem 1.1) that given a \textit{free} homogeneous structure \(M\) (meaning \(M\) is the Fraïssé limit of a class with a stronger form of amalgamation property), then provided \(\Aut(M)\) is a transitive, proper subgroup of \(M\), it is simple. In particular this result encompasses the (previously known) cases of: the random graph [\textit{J. K. Truss}, Math. Proc. Camb. Philos. Soc. 105, No. 2, 223-236 (1989; Zbl 0687.20003)], the random \(K_n\)-free graph [\textit{M. Rubin}, unpublished], and the random tournament [\textit{E. Jaligot}, unpublished]. The methods are borrowed both from descriptive set theory (the automorphism group is a Polish group) and model theory, with an abstract analogue of stability-theoretic independence in the context of Fraïssé limits (though no stability theory is used in the article). The authors ``believe [their] method to have considerable potential for further generalization''.
scientific article; zbMATH DE number 6005095
scientific article