Baily–Borel compactification
Encyclopedia
In mathematics, the Baily–Borel compactification is a compactification
of a quotient of a Hermitian symmetric space
by an arithmetic group
, introduced by .
Compactification (mathematics)
In mathematics, compactification is the process or result of making a topological space compact. The methods of compactification are various, but each is a way of controlling points from "going off to infinity" by in some way adding "points at infinity" or preventing such an "escape".-An...
of a quotient of a Hermitian symmetric space
Hermitian symmetric space
In mathematics, a Hermitian symmetric space is a Kähler manifold M which, as a Riemannian manifold, is a Riemannian symmetric space. Equivalently, M is a Riemannian symmetric space with a parallel complex structure with respect to which the Riemannian metric is Hermitian...
by an arithmetic group
Arithmetic group
In mathematics, an arithmetic group in a linear algebraic group G defined over a number field K is a subgroup Γ of G that is commensurable with G, where O is the ring of integers of K. Here two subgroups A and B of a group are commensurable when their intersection has finite index in each of them...
, introduced by .
Example
- If C is the quotient of the upper half plane by a congruence subgroupCongruence subgroupIn mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible 2x2 integer matrices of determinant 1, such that the off-diagonal entries are even.An importance class of congruence...
of SL2(Z), then the Baily–Borel compactification of C is formed by adding a finite number of cusps to it.