Existence of a dissipated mitotic group

Determine whether there exists a group that is both dissipated—in the sense of a boundedly supported group of bijections of a set X admitting, for each bounded subset X_i, a dissipator t_i with t_i^p(X_i) ∩ X_i = ∅ for all p ≥ 1 and such that the diagonal action f(g) built from the t_i^p-translates of a compactly supported element g lies in the group—and mitotic, meaning that for every finitely generated subgroup H ≤ Γ there exist elements t_1, t_2 ∈ Γ with [H, {}^{t_1}H] = 1 and {}^{t_2}h = h · {}^{t_1}h for all h ∈ H.

Background

Mitotic groups form a notable subclass of binate groups arising from combinatorial group theory, while dissipated groups typically arise as boundedly supported transformation groups in geometric settings (e.g., compactly supported homeomorphisms). Although both classes are binate and each of them independently implies strong cohomological vanishing, their intersection is unclear.

The authors explain that, under certain displacement assumptions (commuting conjugates), a hypothetical group that is both dissipated and mitotic would impose strong constraints on supports of finitely generated subgroups. Despite this, they cannot rule out the existence of such a group, leading to the explicit problem.