Compactification and Discrete Subgroups of Lie Groups
In the study of discrete subgroups of Lie groups, the quotient space of cosets is often a candidate for more subtle compactification to preserve structure at a richer level than just topological.
For example modular curves are compactified by the addition of single points for each cusp, making them Riemann surfaces (and so, since they are compact, algebraic curves). Here the cusps are there for a good reason: the curves parametrize a space of 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 n-dimensional Euclidean space the same questions can be posed, for example about SO(n)\SLn(R)/SLn(Z). 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 compactifications, that can be formed.
Read more about this topic: Compactification (mathematics)
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