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Is it possible to have a structure $T$ in some language which is rigid in $V$, but in a cardinal-preserving extension $T$ is homogeneous (in a suitable sense of the word)?

If this is not possible, is it at least consistent? If not, then what if we remove the requirement that cardinals are preserved?

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  • $\begingroup$ A structure $M$ is homogeneous if, whenever $A \mapsto B$ is a partial elementary map from $M$ to itself, where $|A|, |B| < |M|$, then for all $a \in M$ there is $b \in M$ such that $Aa \mapsto Bb$ is partial elementary. So this in fact compatible with rigidity: say let $M$ be any structure of size $\kappa$ for the language with $\kappa$-many constant symbols. Then $M$ is homogeneous and rigid. You probably want to ask for many automorphisms, rather than homogeneity. $\endgroup$
    – Danielle Ulrich
    Commented Jun 10, 2016 at 21:51

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Yes, it is at least consistent. See:

Fuchs, Gunter, Club degrees of rigidity and almost Kurepa trees. Arch. Math. Logic 52 (2013), no. 1-2, 47–66.

In this paper a (very) rigid Souslin tree $T$ is constructed such that after forcing with $T$ itself, $\aleph_2$-many automorphisms of $T \restriction Lim$ are added. So it suffices to consider the structure $T \restriction Lim$ and force with $T$.

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