Sh:1109

• Paolini, G., & Shelah, S. (2019). Reconstructing structures with the strong small index property up to bi-definability. Fund. Math., 247(1), 25–35. arXiv: 1703.10498 DOI: 10.4064/fm640-9-2018 MR: 3984277
• Abstract:
  Let $\mathbf{K}$ be the class of countable structures $M$ with the strong small index property and locally finite algebraicity, and $\mathbf{K}_*$ the class of $M \in \mathbf{K}$ such that $acl_M(\{ a \}) = \{ a \}$ for every $a \in M$. For homogeneous $M \in \mathbf{K}$, we introduce what we call the expanded group of automorphisms of $M$, and show that it is second-order definable in $Aut(M)$. We use this to prove that for $M, N \in \mathbf{K}_*$, $Aut(M)$ and $Aut(N)$ are isomorphic as abstract groups if and only if $(Aut(M), M)$ and $(Aut(N), N)$ are isomorphic as permutation groups. In particular, we deduce that for $\aleph_0$-categorical structures the combination of strong small index property and no algebraicity implies reconstruction up to bi-definability, in analogy with Rubin’s well-known $\forall \exists$-interpretation technique of [Rubin] Finally, we show that every finite group can be realized as the outer automorphism group of $Aut(M)$ for some countable $\aleph_0$-categorical homogeneous structure $M$ with the strong small index property and no algebraicity.
• Version 2017-04-03_2 (8p) published version (12p)

@article{Sh:1109,
 author = {Paolini, Gianluca and Shelah, Saharon},
 title = {{Reconstructing structures with the strong small index property up to bi-definability}},
 journal = {Fund. Math.},
 fjournal = {Fundamenta Mathematicae},
 volume = {247},
 number = {1},
 year = {2019},
 pages = {25--35},
 issn = {0016-2736},
 mrnumber = {3984277},
 mrclass = {20B27 (03C15 03C35)},
 doi = {10.4064/fm640-9-2018},
 note = {\href{https://arxiv.org/abs/1703.10498}{arXiv: 1703.10498}},
 arxiv_number = {1703.10498}
}