# Cusps for higher dimensional hyperbolic spaces

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:

$$\mathfrak h^n= \{ A \in GL_n(\mathbb R) \, | \, A \text{ is upper triangular and } A_{i,i}>0, \, A_{n,n}=1 \}.$$ 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.

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The correct buzzword here is "symmetric spaces". If $G$ is a semisimple Lie group, then the symmetric space for $G$ is $G / K$, where $K$ is a maximal compact subgroup of $G$ (well-defined up to conjugacy). It's a nice exercise to check that you get the usual upper half-plane by taking $G = SL_2(\mathbb{R})$ and $K = SO_2(\mathbb{R})$. I haven't checked too carefully but it looks very much like your example is the symmetric space for $SL_n(\mathbb{R})$.
Now, a subtle point about cusps for the upper half-plane. It's not the case that "the cusps of $\mathfrak{h}$ are $\mathbb{Q} \cup \infty$". If you take the quotient of $\mathfrak{h}$ by a subgroup $\Gamma$ of $SL_2(\mathbb{R})$ which is commensurable with $SL_2(\mathbb{Z})$, then the "right" way to compactify this is to throw in $\mathbb{Q} \cup \infty$, which is preserved by $\Gamma$. But there are other types of discrete groups acting on $\mathfrak{h}$, and if you want to compactify the quotient by one of these, the set of extra points you have to add will be different (in particular, $\Gamma$ won't necessarily preserve $\mathbb{Q} \cup \infty$). (Silly example: there are discrete groups $\Gamma$ for which the quotient $\Gamma \backslash \mathfrak{h}$ is compact, and there the right set of "cusps" to use is clearly the empty set!). So the definition of "cusp" depends on the additional data of a commensurability class of discrete subgroups.
Now, back to the higher-dimensional case. There is a very rich theory of compactifications of symmetric spaces (or more precisely of locally symmetric spaces, which are the quotients of symmetric spaces by discrete subgroups of $G$). It turns out that if your $G$ is defined over $\mathbb{Q}$, and you want to quotient out by the action of arithmetic subgroups of $G$, then there are canonical good compactifications given in terms of the conjugacy classes of $\mathbb{Q}$-rational parabolic subgroups of $G$. I use the plural because there are two subtly different standard compactifications: the Baily-Borel and Borel-Serre compactifications. Googling the name of either of these will bring up lots of relevant information.