Free groups generated by two unipotent maps

Let A and B be two unipotent elements of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>SU</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">〈</m:mo> <m:mi>A</m:mi> <m:mo>,</m:mo> <m:mi>B</m:mi> <m:mo stretchy="false">〉</m:mo> </m:mrow> </m:math> { A,B} 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">〈</m:mo> <m:mi>A</m:mi> <m:mo>,</m:mo> <m:mi>B</m:mi> <m:mo stretchy="false">〉</m:mo> </m:mrow> </m:math> { A,B} is discrete and free in [J. R. Parker and P. Will, A complex hyperbolic Riley slice, Geom. Topol. 21 2017, 6, 3391–3451].