The automorphism subgroup fixing a strong cut in countable recursively saturated PA models has the small index property, uncountable cofinality, and meagre nontrivial normal subgroups.
Hajnal,The chromatic number of the product of twoℵ 1-chromatic graphs can be countable,Combinatorica
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The automorphism group of countable recursively saturated models of Peano arithmetic and strong cuts
The automorphism subgroup fixing a strong cut in countable recursively saturated PA models has the small index property, uncountable cofinality, and meagre nontrivial normal subgroups.