(2,3,7) triangle group

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In the theory of Riemann surfaces and hyperbolic geometry, the triangle group (2, 3, 7) is particularly important. This importance stems from its connection to Hurwitz surfaces, namely Riemann surfaces of genus g with the largest possible order, 84(g − 1), of its automorphism group.

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[edit] Hyperbolic construction

To construct the triangle group, start with a hyperbolic triangle with angles \pi/2, \;\pi/3,\; \pi/7. Consider then the group generated by reflections in the sides of the triangle. The Fuchsian group defined by the index 2 subgroup consisting of the orientation-preserving isometries is then, by definition, the (2,3,7) triangle group.

[edit] Group-theoretic construction

It has a presentation in terms of a pair of generators, g2 and g3, modulo the following relations:

g_2^2=g_3^3= (g_2g_3)^7=1.

The (2, 3, 7) triangle group admits a presentation in terms of the group of quaternions of norm 1 in a suitable order in a quaternion algebra. More specifically, the triangle group is the quotient of the group of quaternions by its center \pm 1.

Torsion-free normal subgroups of the (2, 3, 7) triangle group are Fuchsian groups associated with Hurwitz surfaces, such as the Klein quartic, Macbeath surface and First Hurwitz triplet.

[edit] Quaternion algebra construction

Let η = 2cos(2π / 7). Note the identity

(2 − η)3 = 7(η − 1)2.

Thus the field \mathbb Q(\eta) is a cubic totally real extension of the rationals. The (2,3,7) hyperbolic triangle group is a subgroup of the group of norm 1 elements in the quaternion algebra generated as an associative algebra by the pair of generators i,j and relations i^2=j^2=\eta, \; ij=-ji. One chooses a suitable Hurwitz quaternion order \mathcal Q_{\mathrm{Hur}} in the quaternion algebra. Here the order QHur is generated by elements

g_2= \frac{1}{\eta}ij

and

g_3=\frac{1}{2}(1+(\eta^2-2)j+(3-\eta^2)ij).

In fact, the order is a free \mathbb Z[\eta]-module over the basis 1,g2,g3,g2g3. Here the generators satisfy the relations

g_2^2=g_3^3= (g_2g_3)^7=-1,

which descend to the appropriate relations in the triangle group, after quotienting by the center.

[edit] Relation to SL(2,R)

Extending the scalars from \mathbb Q (\eta) to \mathbb R (via the standard imbedding), one obtains an isomorphism between the quaternion algebra and the group SL(2,R). Choosing a concrete isomorphism allows one to exhibit the (2,3,7) triangle group as a specific Fuchsian group. However, for many purposes, explicit isomorphisms are unnecessary. Thus, traces of group elements (and hence also translation lengths of hyperbolic elements acting in the upper half-plane, as well as systoles of Fuchsian subgroups) can be calculated by means of the reduced trace in the quaternion algebra, and the formula

tr(\gamma)= 2\cosh(\ell_{\gamma}/2).

[edit] References

  • Elkies, N.: Shimura curve computations. Algorithmic number theory (Portland, OR, 1998), 1–47, Lecture Notes in Computer Science, 1423, Springer, Berlin, 1998. See arXiv:math.NT/0005160