Saturated models

The automorphism group

In a recent paper Nurkhaidarov and Schmerl 2011 proved the following: Let $\kappa$ be regular, uncountable, and such that $\kappa^{<\kappa}=\kappa$. For each completion $T\supseteq PA$ let $M^\kappa_T$ be the saturated model of $T$ of cardinality $\kappa$. There is a set $\mathcal T$ of completions of $PA$, such that $|{\mathcal T}|=2^{\aleph_0}$ and for all $T, T'\in {\mathcal T}$, if $T\not=T'$, then ${\rm Aut}(M^\kappa_T)\not\cong {\rm Aut}(M^\kappa_{T'})$. The following question is left open: Are there $T$ and $T'$ such that $T\not=T'$ and ${\rm Aut}(M^\kappa_T)\cong {\rm Aut}(M^\kappa_{T'})$?

Elementary cuts

A cut in a saturated model is balanced if its upward and downward cofinalities are equal to the cardinality of the model. Since there are continuum theories of pairs $(N,M)$, where $N\models PA$ and $M\models PA$, and $N$ is an elementary end extension of $M$, it follows that for every saturated model $N$ there are continuum non elementarily equivalent, hence nonisomorphic, pairs $(N,M)$, where $M$ is an elementary balanced cut of $N$. For a saturated $N$, what is the maximum number of such pairs?

For an almost complete classification of elementary cuts in saturated models of $PA$ see James Schmerl Elementary cuts in saturated models of Peano arithmetic. Notre Dame J. Form. Log. 53 (2012), no. 1, 1–13,