Abstract
Let A and B be two Heisenberg translations of $$\mathrm{PU}(2,1)$$ . In this paper, we will discuss the groups $$\langle A, B \rangle $$ generated by two non-commuting Heisenberg translations and determine when they are free. The main result of the paper improves an assertion made by Xie et al. (Canad. Math. Bull. 56(4) (2013) 881–889). We also extend the improved result for two Heisenberg translations in $$\mathrm{PSp}(2,1)$$ .
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