A further application of indiscernibles is to show that a consistent first-order theory with infinite models has models with many automorphisms. In particular, every first-order theory $T$ (in a countable vocabulary) possessing an infinite model has a model whose automorphism group has an undecidable first-order theory (in the vocabulary of groups). This result is due to Bludov, Giraudoux, Glass, and Sabbagh.

I believe this result can be extended to to show that every abstract elementary class $A$ which has members of unboundedly large cardinality also has members in every large enough power which possess undecidable automorphism groups. It follows that $A$ has non-rigid models. If $A$ also has rigid models in large enough cardinality, then $A$ is not categorical. This will hold for for classes defined by sentences or theories in some infinitary logics, e.g. $L_{\omega_{1} \omega}$, a most interesting case.

It may be very hard to construct rigid models in an AEC, and these are frequently not absolutely rigid, e.g. above the first $\omega$-Erdos cardinal, their rigidity can be destroyed by forcing. Under $GCH$ or $V=L$, one can neverthless attempt to build rigid models using diamonds $\lozenge_{\kappa}$ to eliminate automorphisms; in this way, the relative consistency of results refuting categoricity conjectures can be approached.