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In mathematics, the modular group is the projective special linear group (,) of 2 × 2 matrices with integer coefficients and determinant 1. The matrices A and − A are identified. The modular group acts on the upper-half of the complex plane by fractional linear transformations , and the name "modular group" comes from the relation to ...
The braid group B 3 is the universal central extension of the modular group. Under this covering, the preimage of the modular group PSL(2, Z) is the braid group on 3 generators, B 3, which is the universal central extension of the modular group. These are lattices inside the relevant algebraic groups, and this corresponds algebraically to the ...
A modular function is a function that is invariant with respect to the modular group, but without the condition that f (z) be holomorphic in the upper half-plane (among other requirements). Instead, modular functions are meromorphic : they are holomorphic on the complement of a set of isolated points, which are poles of the function.
The modular group SL(2, Z) acts on the upper half-plane by fractional linear transformations.The analytic definition of a modular curve involves a choice of a congruence subgroup Γ of SL(2, Z), i.e. a subgroup containing the principal congruence subgroup of level N for some positive integer N, which is defined to be
The infinite general linear group or stable general linear group is the direct limit of the inclusions GL(n, F) → GL(n + 1, F) as the upper left block matrix. It is denoted by either GL(F) or GL(∞, F), and can also be interpreted as invertible infinite matrices which differ from the identity matrix in only finitely many places. [12]
A Fuchsian group Γ corresponds to the orbifold obtained from the quotient of the upper half-plane.By a stacky generalization of Riemann's existence theorem, there is a correspondence between the ring of modular forms of Γ and a particular section ring closely related to the canonical ring of a stacky curve.
Thus their quotient, and therefore j, is a modular function of weight zero, in particular a holomorphic function H → C invariant under the action of SL(2, Z). Quotienting out by its centre { ±I } yields the modular group, which we may identify with the projective special linear group PSL(2, Z).
In mathematics, and more precisely in topology, the mapping class group of a surface, sometimes called the modular group or Teichmüller modular group, is the group of homeomorphisms of the surface viewed up to continuous (in the compact-open topology) deformation.