The automorphism group of countable recursively saturated models of Peano arithmetic and strong cuts

TL;DR

This paper extends the concept of Lascar generic automorphisms to models of Peano arithmetic, analyzing the structure and properties of automorphism subgroups fixing strong cuts in countable recursively saturated models.

Contribution

It introduces a new subgroup of automorphisms fixing strong cuts and proves key properties like small index, uncountable cofinality, and the nature of normal subgroups.

Findings

The subgroup has the small index property.

The cofinality of the subgroup is uncountable.

Nontrivial normal subgroups are meagre, and Z is not a homomorphic image.

Abstract

In this paper, we extend the concept of a Lascar generic automorphism in the setting of models of Peano arithmetic ($\mathrm{PA}$) to the subgroup of the automorphism group of a countable recursively saturated model $\mathcal{M}$ of $\mathrm{PA}$ that fixes pointwise a strong cut $I$ of $\mathcal{M}$, denoted by $(\mathrm{Aut}(\mathcal{M}))_{(I)}$. Then, we prove that: (1) $(\mathrm{Aut}(\mathcal{M}))_{(I)}$ has the small index property. (2) The cofinality of $(\mathrm{Aut}(\mathcal{M}))_{(I)}$ is uncountable. (3) Any nontrivial normal subgroup of $(\mathrm{Aut}(\mathcal{M}))_{(I)}$ is meagre in it. In particular, the infinite cyclic group $\mathbb{Z}$ is not a homomorphic image of $(\mathrm{Aut}(\mathcal{M}))_{(I)}$.