Singularity of mixed identities in oligomorphic automorphism groups

Prove that for every countably infinite relational structure (X,R) whose automorphism group Aut(X,R) is oligomorphic, every mixed identity of Aut(X,R) is singular, i.e., no non‑singular mixed identity holds. Here a mixed identity is a word with constants w in the free product Aut(X,R) * F_r whose associated word map is constantly the identity element of Aut(X,R).

Background

The paper studies mixed identities (word maps with constants that evaluate to the identity on the entire group) for automorphism groups of countable relational structures, with a focus on oligomorphic automorphism groups. The authors prove broad non-existence results for many classes and examples, but formulate a general conjecture asserting that only singular mixed identities can occur in this setting.

They verify the conjecture in numerous examples (e.g., Sym(ω), certain classical groups over finite fields, the Rado graph, free amalgamation classes, and the random poset), and provide partial results for Aut(Q,<). The conjecture encapsulates the overarching open direction motivating the work.