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

We prove the toral rank conjecture of Halperin in some new cases. Our results apply to certain elliptic spaces that have a two-stage Sullivan minimal model, and are obtained by combining new lower bounds for the dimension of the cohomology and new upper bounds for the toral rank. The paper concludes with examples and suggestions for future work.