Non-existence of non‑singular mixed identities for Homeo([0,1])

Prove that the group Homeo([0,1]) of orientation-preserving homeomorphisms of the unit interval admits no non‑singular mixed identities.

Background

Section 5 develops techniques for excluding non‑singular mixed identities in Homeo([0,1]) by leveraging a dense embedding of Aut(Q,<) and cocycle methods for bi‑Lipschitz homeomorphisms. The authors prove strong non‑existence results for certain classes of words, including all one‑variable non‑singular words and words whose content lies in [[F2,F2],[F2,F2]].

They conclude by proposing a broader conjecture that no non‑singular mixed identities exist for Homeo([0,1]) at all, paralleling their interest in Aut(Q,<).