- (2,3,7) triangle group
In the theory of
Riemann surfaces andhyperbolic geometry , thetriangle 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.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.
Group-theoretic construction
It has a presentation in terms of a pair of generators, g_2 and g_3, 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 (see below). 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 surface s, such as theKlein quartic ,Macbeath surface andFirst Hurwitz triplet .Quaternion algebra construction
Let eta=2cos (2pi/7). Note the identity
:2-eta)^3= 7(eta-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 suitableHurwitz quaternion order mathcal Q_{mathrm{Hur in the quaternion algebra. Here the order Q_{mathrm{Hur 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 overthe basis 1,g_2,g_3, g_2g_3. 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.
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 specificFuchsian group . However, for many purposes, explicit isomorphisms are unnecessary. Thus, traces of group elements (and hence also translation lengths of hyperbolic elements acting in theupper 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:operatorname{tr}(gamma)= 2cosh(ell_{gamma}/2).,
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
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