By Calegari D.
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A die is a convex 3–dimensional polyhedron. We can ask under what conditions a die is fair — that is, the probability that the die will land on a given side is 1/n where n is the number of sides. This is a very hard problem to treat in full generality, since it is very hard to calculate these probabilities for a generic polyhedron. But there are certain circumstances under which it is easy to show that these probabilities are all equal; if for any two faces f1 , f2 of a die D there is a symmetry of D to itself taking f1 to f2 then the die is manifestly fair.
If P is a polyhedron in X one of Sn , En , Hn whose dihedral angles between top dimensional faces are all of the form π/mi for integers mi , the group GP generated by reflections in these faces of P acts properly discontinuously on X with fundamental domain P . GP has a subgroup of index 2 consisting of orientation preserving elements, which has as fundamental domain a copy of P and its mirror image P . This follows from a theorem called Poincar´e’s polyhedron theorem. A precise statement and discussion are found in [7].
This description of P SL(2, Z) as a group of automorphisms of a tree gives another way to see that it is isomorphic to Z/2Z ∗ Z/3Z. For a rational point p/q we can consider the straight line l perpendicular to the real axis given by Re(z) = p/q. As this line l moves from ∞ to p/q it crosses through many different triangles of T , and therefore determines a word w in the letters R, L and their inverses. By induction, it is easy to show that the word w is of the form w = Lm1 Rn1 Lm2 Rn2 . . Lmk Rnk where 1 1 1 1 1 p = ...