Free groups generated by two unipotent maps

Abstract

Let A and B be two unipotent elements of SU ⁢ ( 2 , 1 ) {\mathrm{SU}(2,1)} with distinct fixed points. In [S. B. Kalane and J. R. Parker, Free groups generated by two parabolic maps, Math. Z. 303 2023, 1, Paper No. 9], the authors gave several conditions that guarantee the subgroup 〈 A , B 〉 {\langle A,B\rangle} is discrete and free by using Klein’s combination theorem. We will improve their conditions by using a variant of Klein’s combination theorem. With the same arguments and the additional assumption that AB is unipotent, we also extend Parker and Will’s condition that guarantees the subgroup 〈 A , B 〉 {\langle A,B\rangle} is discrete and free in [J. R. Parker and P. Will, A complex hyperbolic Riley slice, Geom. Topol. 21 2017, 6, 3391–3451].