The limit multiplicities and von Neumann dimensions

Given an arithmetic subgroup Γ in a semi-simple Lie group G, the multiplicity of an irreducible representation of G in L^2(Γ\G) is unknown in general.
We observe the multiplicity of any discrete series representation pi of SL (2, R) in L^2 (Γ(n)\SL (2, R)) is close to the von Neumann dimension of pi over the group algebra of Γ(n).
We extend this result to other Lie groups and bounded families of irreducible representations of them.
By applying the trace formulas, we show the multiplicities are exactly the von Neumann dimensions if we take certain towers of descending lattices in some Lie groups.