The examples in the previous section were obtained by applying a theorem of Sunada . Let be a finite group. Call two subgroups and of isospectral if each element of belongs to just as many conjugates of as of . (This is equivalent to requiring that and have the same number of elements in each conjugacy class of .) Sunada's theorem states that if acts on a manifold and and are isospectral subgroups of , then the quotient spaces of by and are isospectral.
The tables in this section show for each of the examples a trio of elements which generate the appropriate , in two distinct permutation representations. The isospectral subgroups and are the point-stabilizers in these two permutation representations.
For the example , the details are as follows. is the group of motions of the hyperbolic plane generated by the reflections in the sides of a triangle whose three angles are . In Conway's orbifold notation (see ), . has a homomorphism onto the finite group (also known as ), the automorphism group of the projective plane of order 2. The generators of act on the points and lines of this plane (with respect to some unspecified numbering of the points and lines) as follows:
where the actions on points and lines are separated by .
The group has two subgroups and of index 7, namely the stabilizers of a point or a line. The preimages and of these two groups in have fundamental regions that consist of 7 copies of the original triangle, glued together as in Figure 2. Each of these is a hexagon of angles , and so each of and is a copy of the reflection group .
The preimage in of the trivial subgroup of is a group of index 168. The quotient of the hyperbolic plane by is a 23-fold cross-surface (that is to say, the connected sum of 23 real projective planes), so that in Conway's orbifold notation . Deforming the metric on this 23-fold cross surface by replacing its hyperbolic triangles by scalene Euclidean triangles yields a cone-manifold whose quotients by and are non-congruent planar isospectral domains.
Tables 1 and 2 display the corresponding information for our other examples.
Table 1: Specifications.
Table 2: Permutations.
Note that the permutations in Table 2 correspond to the neighboring relations in Figure 4. In the propeller example, for instance, the pairs 0, 1 and 2, 5 are neighbors along a dotted line on the left-hand side, and 0, 4 and 2, 3 are neighbors along a dotted line on the right-hand side. Accordingly, we have the permutations a = (0 1)(2 5) / (0 4)(2 3), etc. Similar relations will hold in the other pairs of diagrams if the triangles are properly labelled.