Let {Gr,i} be a sequence of r-generator Kleinian groups acting on . In this paper, we prove that if {Gr,i} satisfies the F-condition, then its algebraic limit group Gr is also a Kleinian group. The existence of a homomorphism from Gr to Gr,i is also proved. These are generalisations of all known corresponding results.

Jørgensen’s famous inequality gives a necessary condition for a subgroup of PSL(2,ℂ) to be discrete. It is also true that if Jørgensen’s inequality holds for every nonelementary two-generator subgroup, the group is discrete. The sufficient condition has been generalized to many settings. In this paper, we continue the work of Wang, Li and Cao (‘Discreteness criteria for Möbius groups acting on ’, Israel J. Math.150 (2005), 357–368) and find three more (infinite) discreteness criteria for groups acting on ; we also correct a linguistic ambiguity of their Theorem 3.3 where one of the necessary conditions might be vacuously fulfilled. The results of this paper are obtained by using known results regarding two-generator subgroups and a careful analysis of the relation among the fixed point sets of various elements of the group.