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In Shimura's Intro to Arithmetic Theory of Automorphic Forms, he defines a cusp of a Fuchsian group $\Gamma$ as a point $s \in \mathbb{R} \cup \{ \infty \}$ that is fixed by a parabolic element of $\Gamma$.

I know the definition of the width of a cusp in the case $\Gamma = SL_2(\mathbb{Z})$ or a congruence subgroup (namely, it is the only positive integer $h$ such that $\rho^{-1} \overline{\Gamma_s} \rho$ is generated by the matrix $\begin{pmatrix} 1 & h \\\\ 0 & 1 \end{pmatrix}$, where $\rho$ is any matrix in $SL_2(\mathbb{Z})$ such that $\rho(s) = \infty$) but I was wondering if we can also define a notion of cusp width for any Fuchsian group.

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2 Answers 2

up vote 9 down vote accepted

For subgroups $ \Gamma \subset SL_2(\mathbb Z)$, every cusp is a covering space of the single cusp for the quotient of the upper half plane by $SL_2(\mathbb Z)$ (that is, the (2,3,infinity orbifold), and the index of the covering is a natural concept that equals the width. There is no such structure in general: in fact, (as is well-known and I'm sure is described in Shimura's book) subgroups of $SL_2(\mathbb Z)$ are conjugate (within $GL_2(\mathbb Q)$) to other subgroups where this index changes: for example, let $\Gamma$ be the subgroup of $SL_2(\mathbb Z)$ that when conjugated by the diagonal matrix with diagonal entries $2,1$ is still integral.

There is another, related concept, however, that is universally applicable: the set of areas of maximal horoball neighborhoods of cusps. For any cusp in the quotient of upper half space by a Fuchsian group, there is some maximal neighborhood of the cusp of the form horoball mod parabolic subgroup (stabilizer of cusp) that is embedded in the quotient. Its area (which equals the length of its boundary curve) is an invariant. If the quotient orbifold has $k$ cusps, then there is moreover a convex subset of $\mathbb R^k$ consisting of the set of $k$-tuples of logs of areas of these horoball neighborhoods such that the union is embedded. It is bounded by hyperplanes of the form $\log(A_i) < C_i$ and $\log(A_i) + \log(A_j) < K_{i,j}$.

For the 3-dimensional generalization of this (having to do with discrete subgroups of $PSL(2,\mathbb C)$ act on hyperbolic 3-space, Jeff Weeks' program snappea is very good for getting a sense of how these horoball neighborhoods interact: you can see the pictures change as you move sliders. Unfortunately the best version of the original snappea (you can easily find it by googling) only runs on obsolete versions of the macintosh operating system, but there is a modern version, SnapPy, modernized using Python by Marc Culler and Nathan Dunfield, that will give the same information, but requires more keyboard and less GUI interaction. The geometry of the maximal horoball neighborhoods of cusps has considerable significance for 3-dimensional topology, and has been studied quite a lot.

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The 1.3.4 version of SnapPy has a much better cusp viewer, complete with sliders similar to Weeks' GUI. I believe that Culler and Dunfield are eager to get feedback and feature requests (see also math.uic.edu/t3m/SnapPy/todo.html). –  Sam Nead Mar 29 '11 at 11:59
    
Thanks for the useful reply! I had never heard about these maximal horoballs and their areas though. Can you point me to a reference? –  expmat Mar 29 '11 at 13:44

You can define it as the length of the longest embedded horocycle on the quotient surface, which is often a useful thing to look at, but this really depends on what you need it for...

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