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Cusp neighborhood

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(Redirected from Cusp neighbourhood) Neighborhood of a singularity of cusp type
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In mathematics, a cusp neighborhood is defined as a set of points near a cusp singularity.

Cusp neighborhood for a Riemann surface

The cusp neighborhood for a hyperbolic Riemann surface can be defined in terms of its Fuchsian model.

Suppose that the Fuchsian group G contains a parabolic element g. For example, the element t ∈ SL(2,Z) where

t ( z ) = ( 1 1 0 1 ) : z = 1 ⋅ z + 1 0 ⋅ z + 1 = z + 1 {\displaystyle t(z)={\begin{pmatrix}1&1\\0&1\end{pmatrix}}:z={\frac {1\cdot z+1}{0\cdot z+1}}=z+1}

is a parabolic element. Note that all parabolic elements of SL(2,C) are conjugate to this element. That is, if g ∈ SL(2,Z) is parabolic, then g = h − 1 t h {\displaystyle g=h^{-1}th} for some h ∈ SL(2,Z).

The set

U = { z ∈ H : ℑ z > 1 } {\displaystyle U=\{z\in \mathbf {H} :\Im z>1\}}

where H is the upper half-plane has

γ ( U ) ∩ U = ∅ {\displaystyle \gamma (U)\cap U=\emptyset }

for any γ ∈ G − ⟨ g ⟩ {\displaystyle \gamma \in G-\langle g\rangle } where ⟨ g ⟩ {\displaystyle \langle g\rangle } is understood to mean the 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

E = U / ⟨ g ⟩ {\displaystyle E=U/\langle g\rangle } .

Here, E is called the neighborhood of the cusp corresponding to g.

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

{ z ∈ H : | z | > 1 , | Re ( z ) | < 1 2 } {\displaystyle \left\{z\in H:\left|z\right|>1,\,\left|\,{\mbox{Re}}(z)\,\right|<{\frac {1}{2}}\right\}}

of the modular group, as would be appropriate for the choice of T as the parabolic element. When integrated over the volume element

d μ = d x d y y 2 {\displaystyle d\mu ={\frac {dxdy}{y^{2}}}}

the result is trivially 1. Areas of all cusp neighborhoods are equal to this, by the invariance of the area under conjugation.

See also

References

  1. Fujikawa, Ege; Shiga, Hiroshige; Taniguchi, Masahiko (2004). "On the action of the mapping class group for Riemann surfaces of infinite type". Journal of the Mathematical Society of Japan. 56 (4): 1069–1086. doi:10.2969/jmsj/1190905449.
  2. Basmajian, Ara (1992). "Generalizing the hyperbolic collar lemma". Bulletin of the American Mathematical Society. 27 (1): 154–158. arXiv:math/9207211. doi:10.1090/S0273-0979-1992-00298-7. ISSN 0273-0979.
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