Reconstructing structures with the strong small index property up to bi-definability

Gianluca Paolini, Saharon Shelah Fundamenta Mathematicae MSC: 20B27, 03C35, 03C15. DOI: 10.4064/fm640-9-2018 Published online: 12 April 2019

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 $\mathop {\rm acl}_M(\{ a \}) = \{ a \}$ 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 $\mathop {\rm Aut}(M)$. We use this to prove that for $M, N \in \mathbf {K}_*$, $\mathop {\rm Aut}(M)$ and $\mathop {\rm Aut}(N)$ are isomorphic as abstract groups if and only if $(\mathop {\rm Aut}(M), M)$ and $(\mathop {\rm Aut}(N), N)$ are isomorphic as permutation groups. In particular, we deduce that for $\aleph _0$-categorical structures the combination of the strong small index property and no algebraicity implies reconstruction up to bi-definability, in analogy with Rubin’s (1994) well-known $\forall \exists $-interpretation technique. Finally, we show that every finite group can be realized as the outer automorphism group of $\mathop {\rm Aut}(M)$ for some countable $\aleph _0$-categorical homogeneous structure $M$ with the strong small index property and no algebraicity.

Authors

  • Gianluca PaoliniDepartment of Mathematics “Giuseppe Peano”
    University of Torino
    Via Carlo Alberto 10
    10123 Torino, Italy
    e-mail
  • Saharon ShelahEinstein Institute of Mathematics
    The Hebrew University of Jerusalem
    Edmond J. Safra Campus
    Givat Ram, 9190401, Jerusalem
    and
    Department of Mathematics
    Rutgers University
    The State University of New Jersey
    Hill Center–Busch Campus 110
    Frelinghuysen Road
    Piscataway, NJ 08854-8019, U.S.A.
    e-mail

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