Coding in the automorphism group of a computably categorical structure

Using new techniques for controlling the categoricity spectrum of a structure, we construct a structure with degree of categoricity but infinite spectral dimension, answering a question of Bazhenov, Kalimulin and Yamaleev. Using the same techniques, we construct a computably categorical structure of non-computable Scott rank, and a structure of computable dimension 2 such that there is no hyperarithmetic isomorphism between the two copies