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| {{Unreferenced|date=October 2008}}
| | 45 yr old Podiatrist Cletus from Greater Napanee, has interests for example 4 wheeling, diet and maintain a journal. Last month just visited Hosios Loukas and Nea Moni of Chios. |
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| In [[mathematics]], a '''cusp neighborhood''' is defined as a set of points near a [[cusp (singularity)|cusp]].
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| ==Cusp neighborhood for a Riemann surface==
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| The cusp neighborhood for a hyperbolic [[Riemann surface]] can be defined in terms of its [[Fuchsian model]].
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| Suppose that the [[Fuchsian group]] ''G'' contains a [[parabolic element]] g. For example, the element ''t'' ∈ SL(2,'''Z''') where
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| :<math>t(z)=\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}:z = \frac{1\cdot z+1}{0 \cdot z + 1} = z+1</math>
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| is a parabolic element. Note that all parabolic elements of SL(2,'''C''') are [[conjugacy class|conjugate]] to this element. That is, if ''g'' ∈ SL(2,'''Z''') is parabolic, then <math>g=h^{-1}th</math> for some ''h'' ∈ SL(2,'''Z''').
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| The set
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| :<math>U=\{ z \in \mathbf{H} : \Im z > 1 \} </math>
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| where '''H''' is the [[upper half-plane]] has
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| :<math>\gamma(U) \cap U = \emptyset</math>
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| for any <math>\gamma \in G - \langle g \rangle </math> where <math>\langle g \rangle</math> is understood to mean the [[group (mathematics)|group]] generated by ''g''. That is, γ acts [[properly discontinuously]] on ''U''. Because of this, it can be seen that the projection of ''U'' onto '''H'''/''G'' is thus
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| :<math>E = U/ \langle g \rangle</math>.
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| Here, ''E'' is called the '''neighborhood of the cusp corresponding to g'''.
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| Note that the hyperbolic area of ''E'' is exactly 1, when computed using the canonical [[Poincaré metric]]. This is most easily seen by example: consider the intersection of ''U'' defined above with the [[fundamental domain]]
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| :<math>\left\{ z \in H: \left| z \right| > 1,\, \left| \,\mbox{Re}(z) \,\right| < \frac{1}{2} \right\}</math>
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| of the [[modular group]], as would be appropriate for the choice of ''T'' as the parabolic element. When integrated over the [[volume element]]
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| :<math>d\mu=\frac{dxdy}{y^2}</math>
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| the result is trivially 1. Areas of all cusp neighborhoods are equal to this, by the invariance of the area under conjugation.
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| [[Category:Hyperbolic geometry]]
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| [[Category:Riemann surfaces]]
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45 yr old Podiatrist Cletus from Greater Napanee, has interests for example 4 wheeling, diet and maintain a journal. Last month just visited Hosios Loukas and Nea Moni of Chios.