Elementarity of Subgroups and Complexity of Theories for Profinite Groups

TL;DR

This paper investigates the logical complexity of profinite subgroups of the infinite symmetric group using tree representations, establishing bounds on their theories and exploring elementary subgroups related to Turing degrees.

Contribution

It introduces a novel approach using tree presentations to analyze the complexity of theories of profinite groups and characterizes elementary subgroups based on Turing degrees.

Findings

Bounds on the existential theories of profinite subgroups are established.

A subclass of profinite groups with orbit independence has a bounded first-order theory.

A constructed example shows a subgroup of computable elements not elementary for existential formulas.

Abstract

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.