Elementarity of Subgroups and Complexity of Theories for Profinite Groups

Although $S_\infty$ (the group of all permutations of $\mathbb{N}$) is size continuum, both it and its closed subgroups can be presented as the set of paths through a countable tree. The subgroups of $S_\infty$ that can be presented this way with finite branching trees are exactly the profinite ones. We use these tree presentations to find upper bounds on the complexity of the existential theories of profinite subgroups of $S_\infty$, as well as to prove sharpness for these bounds. These complexity results enable us to distinguish a simple subclass of profinite groups, those with \emph{orbit independence}, for which we find an upper bound on the complexity of the entire first order theory. Additionally, given a profinite subgroup $G$ of $S_\infty$ and a Turing ideal $I$ we define $G_I$ to be the set of elements in $G$ whose Turing degree lies in $I$. We examine to what extent and under what conditions $G_I$ will be an elementary subgroup of $G$. In particular, we construct a profinite group whose subgroup of computable elements is not elementary even for existential formulas.