On algebraic convergence of non-elementary discrete subgroups of SL(2, $\mathbb{Q}_p$ )

In the Kelinian groups, the study of the algebraic convergence of the sequence of the discrete subgroups is a very important topic, since the algebraic convergence of the sequence of the discrete subgroups can be applied to study the deformations the manifolds and the Hausdorff dimension of the limit sets of the discrete subgroups. With the rapid developments of the p-adic Lie groups and the algebraic dynamical systems, it is very important to study the topics of algebraic convergence of the p-adic discrete subgroups. In this paper, we discuss the algebraic convergence of a sequence { G n, r } of r-generator non-elementary discretesubgroups of PSL(2, $\mathbb{Q}_p$ ) by use of the Jorgensen inequalities in PSL(2, $\mathbb{Q}_p$ ) and the subgroups of PSL(2, $\mathbb{Q}_p$ ) acting isometrically on the hyperbolic Berkovich space. We prove that a sequence { G n , r } of r-generator non-elementary discretesubgroups of PSL(2, $\mathbb{Q}_p$ ) converges to a non-elementary discrete subgroup of PSL(2, $\mathbb{Q}_p$ ) algebraically.