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

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 $(\mathrm{Aut}(\mathcal{M}){)}_{(I)}$.