Do mitotic groups have commuting cyclic conjugates?

Establish whether every mitotic group has commuting cyclic conjugates; explicitly, determine whether for every finitely generated subgroup H ≤ Γ of a mitotic group Γ there exist t ∈ Γ and n ∈ ℕ, n ≥ 2, or n = ∞, such that [H, {}^{t^p}H] = 1 for 1 ≤ p < n and [H, t^n] = 1.

Background

Commuting cyclic conjugates is a displacement property that implies X_sep-bounded acyclicity. Mitotic groups, introduced by Baumslag–Dyer–Heller, are a subclass of binate groups with strong algebraic displacement features. Whether these algebraic features suffice to guarantee the dynamical organization required by commuting cyclic conjugates remains unclear.

The authors highlight this as a central unresolved issue in relating classical algebraic displacement techniques (mitotic) to newer, dynamical displacement properties that control bounded cohomology with separable coefficients.