Lectures on Shimura Curves 2: General Theory of Fuchsian Groups
نویسنده
چکیده
Let R be any commutative ring. Then by GLN (R) we mean the group of all N × N matrices M with entries in R, and which are invertible: det(M) ∈ R×. The determininant map gives a homomorphism of groups which is easily seen to be surjective: defining the kernel to be SLN (R), we get an extension of groups The center Z of GLN (R) consists of the scalar matrices R× · IN , so as a group isomorphic to R×. By definition, PGLN (R) = GLN (R)/Z. The determinant map factors through to give a surjective homomorphism PGLN (R) → R×/ det(Z) = R×/R×2, whose kernel we call PSLN (R). We summarize the situation with the following diagram:
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