We study several problems about $\mathcal{P}(n)$, the symmetric space associated with the real Lie group $SL(n,\mathbb{R})$. We endow the symmetric space $\mathcal{P}(n)$ with an $SL(n,\mathbb{R})$-invariant premetric proposed by Selberg as a substitute for the Riemannian distance. The problems addressed in this study are linked to an algorithm designed to determine generalized geometric finiteness for subgroups of $SL(n,\mathbb{R})$, similar to the algorithm proposed by Riley in hyperbolic spaces based on Poincar