Editing Compactification (mathematics) (section)

== Compactification and discrete subgroups of Lie groups ==

In the study of [[discrete space|discrete]] subgroups of [[Lie group]]s, the [[Quotient space (topology)|quotient space]] of [[coset]]s is often a candidate for more subtle '''compactification''' to preserve structure at a richer level than just topological.

For example, [[modular curve]]s are compactified by the addition of single points for each [[cusp (singularity)|cusp]], making them [[Riemann surface]]s (and so, since they are compact, [[algebraic curve]]s). Here the cusps are there for a good reason: the curves parametrize a space of [[lattice (group)|lattices]], and those lattices can degenerate ('go off to infinity'), often in a number of ways (taking into account some auxiliary structure of ''level''). The cusps stand in for those different 'directions to infinity'.

That is all for lattices in the plane. In {{math|''n''}}-dimensional [[Euclidean space]] the same questions can be posed, for example about <math>\text{SO}(n) \setminus \text{SL}_n(\textbf{R}) / \text{SL}_n(\textbf{Z}).</math> This is harder to compactify. There are a variety of compactifications, such as the [[Borel–Serre compactification]], the [[reductive Borel–Serre compactification]], and the [[Satake compactification]]s, that can be formed.