The small index property of automorphism groups of ab-initio generic structures

Suppose $M$ is a countable ab-initio (uncollapsed) generic structure which is obtained from a pre-dimension function with rational coefficients. We show that if $H$ is a subgroup of $\mbox{Aut}\left(M\right)$ with $\left[\mbox{Aut}\left(M\right):H\right]<2^{\aleph_{0}}$, then there exists a finite set $A\subseteq M$ such that $\mbox{Aut}_{A}\left(M\right)\subseteq H$. This shows that $\mbox{Aut}\left(M\right)$ has the small index property.

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