Take your favourite model of the hyperbolic plane $\mathfrak h$. It is well known how to define cusps, and we know those are $\mathbb Q \cup \{ \infty \}$.
Now pick a higher dimensional hyperbolic space. In particular, I am interested in the following "kind" of hyperbolic space:
\begin{equation} \mathfrak h^n= \{ A \in GL_n(\mathbb R) \, | \, A \text{ is upper triangular and } A_{i,i}>0, \, A_{n,n}=1 \}. \end{equation} This can be viewed as an hyperbolic space by considering each element as a product of a diagonal matrix $y$ with positive eigenvalue, and $n$-th eigenvalue $1$, and a unipotent upper triangular matrix $x$, just like by using the upper-plane model we can define $\mathfrak h$ as the set of matrices $\begin{pmatrix} y & x // 0 & 1 \end{pmatrix}$ with $y>0$.
Now comes my question: what are the cusps in this higher dimensional hyperbolic space? Is there a natural generalization, either by using geometric or algebraic properties of cusps in the usual hyperbolic plane?
Motivation: After studying modular forms and in particular Maass forms, in the $2$-dimensional case (i.e. for congruence subgroups of $SL_2(\mathbb Z)$) I am now studying the more general $n$-dimensional case. While there is still a notion of cusp form, it is only defined in a very implicit way, by introducing some integral and saying that cusp forms are the ones such that the integral vanish identically. I will add more details in case they are deemed to be relevant.