Automorphisms groups of countable recursively saturated models

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Generic automorphisms and the small index property

Lascar [1] showed that countable arithmetically saturated models of $PA$ have generic automorphisms and used them to show that automorphism groups of these models have the small index property.

Do automorphism groups of countable recursively saturated, but nor arithmetically saturated models of $PA$ have generic automorphisms? Do they have the small index property?

It was shown in [2] that every open subgroup of the automorphism group of a countable recursively saturated model of $PA$ extends to a maximal subgroup. Is the same true for countable models which are recursively saturated, but not arithmetically saturated?


How many automorphism groups are there?

By a result of [3], if $M$ and $N$ are countable arithmetically saturated models of the same completion of $PA$, then $M\cong N$ iff ${\rm Aut}(M)\cong {\rm Aut}(N)$. It is open whether the same is true for recursively saturated models which are not arithmetically saturated. It is even open whether there are nonisomorphic countable recursively saturated models whose automorphism groups are isomorphic.

Nurkhaidarov [4] proved that there is a set $\mathcal T$ of completions of $PA$, such that $|{\mathcal T}|=4$ and for all $T, T'\in {\mathcal T}$, if $T\not=T'$, and $M\models T$ and $M'\models T'$ are countable and arithmetically saturated, then ${\rm Aut}(M)\not\cong{\rm Aut}(M')$. Surely, there must be more than four such completions. How many are there?


Normal subgroups

Let $M$ be a countable recursively saturated model of $PA$ and let $G$ be the automorphism group of $M$. By a theorem of Kaye [5] every closed normal subgroup of $G$ is $G_{(I)}$, the pointwise stabilizer of an invariant initial segment $I$ of $M$. If $I$ is an invariant initial segment of $M$, then the group $G_{(>I)}$ of those automorphisms that fix pointwise an initial segment $J>I$ is also a normal subgroup, whose closure in $G$ is $G_{(I)}$.


Problem: Does $G$ have any other normal subgroups?

Recursively saturated automorphisms

Let $M$ be a countable recursively saturated model of $PA$ and let $G$ be the automorphism group of $M$ and let ${\rm RS}(M)$ be the subgroup of $G$ generated by $f\in G$ such that $(M,f)$ is recursively saturated. ${\rm RS}(M)$ is a normal subgroup of $G$, and every $f\in {\rm RS}(M)$ fixes pointwise a nonstandard initial segment of $M$ [6].


Problem: Is ${\rm RS}(M)$ the group of those $f\in G$ that fix pointwise a nonstandard initial segment of $M$?


Th(Aut(M))

If $M$ is a countable, recursively saturated model of PA, then as shown in [7] and [8], ${\rm Th}({\rm Aut}(M))$ is undecidable. What more can be said about its Turing degree?



References

  1. Daniel Lascar. The small index property and recursively saturated models of Peano arithmetic. Automorphisms of first-order structures, pp. 281--292, New York, 1994. MR   bibtex
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  3. Roman Kossak and James H. Schmerl. The automorphism group of an arithmetically saturated model of Peano arithmetic. J. London Math. Soc. (2) 52(2):235--244, 1995. www   DOI   MR   bibtex
  4. Ermek S. Nurkhaidarov. Automorphism groups of arithmetically saturated models. J. Symbolic Logic 71(1):203--216, 2006. www   DOI   MR   bibtex
  5. Richard Kaye. A Galois correspondence for countable recursively saturated models of Peano arithmetic. Automorphisms of first-order structures, pp. 293--312, New York, 1994. MR   bibtex
  6. Richard Kaye, Roman Kossak and Henryk Kotlarski. Automorphisms of recursively saturated models of arithmetic. Ann. Pure Appl. Logic 55(1):67--99, 1991. www   DOI   MR   bibtex
  7. James H. Schmerl. Automorphism groups of models of Peano arithmetic. J. Symbolic Logic 67(4):1249--1264, 2002. www   DOI   MR   bibtex
  8. Error: entry with key = schmerl2012:theAutomorphism does not exist
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