For what it is worth, Ive computed some more terms in the iguanodon series. Here they are

By construction, the n-th iguanodon group (corresponding to the n-th Farey sequence) is a subgroup of the alternating group on its (half)legs. Hence to prove that all remaining iguanodons are alternating groups boils down to proving that they are sufficiently transitive, for example, by showing that there are permutations of certain cycle-types in the group. Im sure any grouptheorist can crack this problem over lunch, so if you did please drop a comment.

Clearly, I didnt do the calculations in the archaic way of the previous post (as depicted on the left) which consisted in adding a pair of new legs at the proper place in the spine for every new Farey number, write down the two generating permutations, giving them to GAP and check simplicity and the isomorphism type.

Instead I used a nice SAGE-package to compute with Farey-symbols written by Chris Kurth and available from his website. As this package is a good tool to experiment hunting for other dinosaur-series of simple groups coming from series of Farey-symbols, Ill include the details for (the example used to outline the construction of the Iguanodon-series ).

First we need to have the n-th Farey-sequence . There are several short Python programs around to do this, for example this one from the Python-Cookbook. Save it to your sage-directory and name it fareyseq.py and load it into sage via load fareyseq.py. Then typing farey(3) to the sage-prompt spits back

sage: farey(3)
[(1, 3), (1, 2), (2, 3)]

That is, 0 and 1 are not included and Farey-numbers are represented by numerator-denominator couples. The iguanodon-series uses the Fareys upto 1/2, identifies the edges connecting 0 and 1 to and makes all other intervals odd. That is, the corresponding Farey symbol for F(3) is

(to add to the confusion, I denote odd intervals by a black-bullet whereas in Kulkarni’s paper they are white…) Anyway, get Kurth’s kfarey-package and save the folder as kfarey in your sage-folder. Kurth uses the following notation for Farey-symbols

The Farey Symbol is a list [a,b,p] where 
a is a list of numerators, b a list of denominators, and p the pairing 
information. If x[i]=a[i]/b[i]: 
inf x0 x1 x2 … xn inf 
  p0 p1 p2 … pn pn+1 
So p[i] is the pairing of the side between x[i-1] and x[i]. The p[i]’s can be 
positive integers, indicating pairing between sides, or -2 or -3, meaning 
an even or odd pairing respectively.

The above Farey-symbol is therefore represented as [[0,1,1,1],[1,3,2,1],[1,-3,-3,-3,1]]. The kfarey-function LRCosetRep(F) returns two permutations L and R giving the permutation action of the two generators of the modular group

on the half-legs of the inguanodon (the dessin corresponding to the Farey-symbol). Here’s the sage transcript

sage: load kfarey/farey.sage
sage: load kfarey/conggroups.sage
sage: load kfarey/LR.sage
sage: ig3=[[0,1,1,1],[1,3,2,1],[1,-3,-3,-3,1]]
sage: LRCosetRep(ig3)
[(1,2,3,9,10,11,6,7,8,4,5)(12), (1,8,4,2,11,6,3,12,10,7,5)(9)]

Giving these two generators to GAP one verifies that they indeed generate

gap> ig3:=Group((1,2,3,9,10,11,6,7,8,4,5)(12), (1,8,4,2,11,6,3,12,10,7,5)(9));
Group([ (1,2,3,9,10,11,6,7,8,4,5), (1,8,4,2,11,6,3,12,10,7,5) ])
gap> IsSimpleGroup(ig3);
true
gap> IsomorphismTypeInfoFiniteSimpleGroup(ig3);
rec( series := “Spor”, name := “M(12)” )

kfarey has plenty of other useful functions. One can even create an .eps file of the fundamental domain specified by the subgroup of the modular group encoded by the Farey symbol using MakeEpsFile(F). For the above example it returns the picture on the right. Not quite as nice as the one on the left, but surely a lot easier to create.

Here the details of the iguanodon series. Start with the Farey sequence of order n which is the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size. Here are the first eight Fareys

F(1) = {0⁄1, 1⁄1}
F(2) = {0⁄1, 1⁄2, 1⁄1}
F(3) = {0⁄1, 1⁄3, 1⁄2, 2⁄3, 1⁄1}
F(4) = {0⁄1, 1⁄4, 1⁄3, 1⁄2, 2⁄3, 3⁄4, 1⁄1}
F(5) = {0⁄1, 1⁄5, 1⁄4, 1⁄3, 2⁄5, 1⁄2, 3⁄5, 2⁄3, 3⁄4, 4⁄5, 1⁄1}
F(6) = {0⁄1, 1⁄6, 1⁄5, 1⁄4, 1⁄3, 2⁄5, 1⁄2, 3⁄5, 2⁄3, 3⁄4, 4⁄5, 5⁄6, 1⁄1}
F(7) = {0⁄1, 1⁄7, 1⁄6, 1⁄5, 1⁄4, 2⁄7, 1⁄3, 2⁄5, 3⁄7, 1⁄2, 4⁄7, 3⁄5, 2⁄3, 5⁄7, 3⁄4, 4⁄5, 5⁄6, 6⁄7, 1⁄1}
F(8) = {0⁄1, 1⁄8, 1⁄7, 1⁄6, 1⁄5, 1⁄4, 2⁄7, 1⁄3, 3⁄8, 2⁄5, 3⁄7, 1⁄2, 4⁄7, 3⁄5, 5⁄8, 2⁄3, 5⁄7, 3⁄4, 4⁄5, 5⁄6, 6⁄7, 7⁄8, 1⁄1}

Farey sequences have plenty of mysterious properties. For example, in 1924 J. Franel and Edmund Landau proved that an asymptotic density result about Farey sequences is equivalent to the Riemann hypothesis. More precisely, let a(n) be the number of terms in the Farey sequence F(n) (that is, a(1)=2,a(2)=3,…,a(8)=23 etc. This is sequence A005728 in the online integer sequences catalog). Let denote the j-th term in F(n), then the following conjecture is equivalent to the Riemann hypothesis

For every there is a constant C depending on such that

when n goes to infinity. Anyway, let us continue our construction. Farey sequences are clearly symmetric around 1/2 so let us just take half of them, so we jump to 1 when we have reached 1/2. Let us extend this halved Farey on both sides with and call it the modified Farey sequence f(n). For example,

Now consider the Farey code in which we identify the two sides connected to and mark two consecutive Farey numbers as

That is, the Farey code associated to the modified sequence f(3) is

Recall from earlier that to a Farey-code we can associate a special polygon by first taking the hyperbolic convex hull of all the terms in the sequence (the region bounded by the vertical lines and the bottom red circles in the picture on the left) and adding to it for each odd interval the triangle just outside the convex hull consisting of two odd edges in the Dedekind tessellation (then we obtain the region bounded by the black geodesics for the sequence f(3)).

Next, we can associate to this special polygon a cuboid tree diagram by considering all even and odd vertices on the boundary (which are tinted red, respectively blue) together with all odd vertices in the interior of the special polygon. These are indicated in the left picture below. If we connect these vertices with the geodesics in the polygon we get a cuboid tree diagram. The obtained cuboid tree diagram is depicted on the right below.

Finally, identifying the red points (as they lie on geodesics connected to which are identified in the Farey code), adding even points on the remaining geodesics and numbering the obtained half-lines we obtain the dessin d’enfant given on the left hand side. To such a dessin we can associate its monodromy group which is a permutation group on the half-lines generated by an order two element indicating which half-lines make up a line and an order three element indicating which half-lines one encounters by walking counter-clockwise around a three-valent vertex. For the dessin on the left the group is therefore the subgroup of generated by the elements

and a verification with GAP tells us that this group is the sporadic Mathieu group . This concludes the description of the second member of the Iguanodon series. If you like to check that the first 8 iguanodons are indeed the simple groups

the following dissection of the Iguanodon may prove useful

Bruce Westbury has a page on recent work on series of Lie groups including exceptional groups. Moreover, he did put his slides of a recent talk (probably at MPI) online.

Probably, someone considered a similar problem for simple groups. Are there natural constructions leading to a series of finite simple groups including some sporadic groups as special members ? In particular, does the following sequence appear somewhere ?

Here, is the simple group of order 168 (the automorphism group of the Klein quartic), and are the sporadic Mathieu groups and the are the alternating simple groups.

I’ve stumbled upon this series playing around with Farey sequences and their associated ‘dessins d’enfants’ (I’ll come back to the details of the construction another time) and have dubbed this sequence the Iguanodon series because the shape of the doodle leading to its first few terms

reminded me of the Iguanodons of Bernissart (btw. this sketch outlines the construction to the experts). Conjecturally, all groups appearing in this sequence are simple and probably all of them (except for the first few) will be alternating.

I did verify that none of the known low-dimensional permutation representations of other sporadic groups appear in the series. However, there are plenty of similar sequences one can construct from the Farey sequences, and it would be nice if one of them would contain the Conway group . (to be continued)