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In mathematics, the modular group is the projective special linear group (,) of matrices with integer coefficients and determinant, such that the matrices and are identified. The modular group acts on the upper-half of the complex plane by linear fractional transformations .
In mathematics, a modular invariant of a group is an invariant of a finite group acting on a vector space of positive characteristic (usually dividing the order of the group). The study of modular invariants was originated in about 1914 by Dickson (2004) .
The projective special linear group, PSL, is defined analogously, as the induced action of the special linear group on the associated projective space. Explicitly: PSL(V) = SL(V) / SZ(V) where SL(V) is the special linear group over V and SZ(V) is the subgroup of scalar transformations with unit determinant.
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]
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 ...
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).
Modular form theory is a special case of the more general theory of automorphic forms, which are functions defined on Lie groups that transform nicely with respect to the action of certain discrete subgroups, generalizing the example of the modular group () ().
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.
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