For what it is worth, Ive computed some more terms in the iguanodon series. Here they are
By construction, the n-th iguanodon group
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
First we need to have the n-th Farey-sequence
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
[tex]\xymatrix{\infty \ar@{-}[r]_{1} & 0 \ar@{-}[r]_{\bullet} & \frac{1}{3} \ar@{-}[r]_{\bullet} & \frac{1}{2} \ar@{-}[r]_{\bullet} & 1 \ar@{-}[r]_{1} & \infty}[/tex]
(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.
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