Fractals/Mathematics/group/Basilica group

Basilica group is :
 * group defined by automatum
 * the iterated monodromy group of the polynomial $$z^2-1$$
 * related with Basilica Julia set : "the scaling limit of the Schreier graphs of its action on level n of T is the basilica"

Computation
The critical points of the polynomial $$z^2-1$$ are $$\infty$$ and $$0$$.

The postcritical set is $$P = \left \{ 0, -1, \infty \right \} $$

FR
predefined by FR package of GAP CAS. Here BinaryKneadingGroup("1") is BasilicaGroup.

gap> BinaryKneadingGroup(1/3)=BasilicaGroup; true

or :

gap> B := FRGroup("a=<1,b>(1,2)","b=<1,a>",IsFRMealyElement);  gap> AssignGeneratorVariables(B); gap> B=BasilicaGroup; true
 * 1) I Assigned the global variables [ "a", "b" ]
 * 1) I \=: converting second argument to FR element
 * 2) I \<: converting second argument to FR element
 * 3) I \<: converting second argument to FR element
 * 4) I \=: converting second argument to FR element
 * 5) I \=: converting second argument to FR element
 * 6) I \<: converting second argument to FR element
 * 7) I \<: converting second argument to FR element
 * 8) I \=: converting second argument to FR element
 * 9) I \=: converting first argument to FR element
 * 10) I \=: converting first argument to FR element
 * 11) I \=: converting first argument to FR element
 * 12) I \=: converting first argument to FR element
 * 13) I \=: converting first argument to FR element
 * 14) I \=: converting first argument to FR element
 * 15) I \=: converting first argument to FR element
 * 16) I \=: converting first argument to FR element

gap> Size(BasilicaGroup); infinity gap> GeneratorsOfGroup(BasilicaGroup); [ a, b ] gap> Alphabet(BasilicaGroup); [ 1, 2 ] gap> KnownAttributesOfObject(BasilicaGroup); [ "Name", "Representative", "OneImmutable", "GeneratorsOfMagma", "GeneratorsOfMagmaWithInverses", "MultiplicativeNeutralElement", "UnderlyingFRMachine", "Correspondence", "AlphabetOfFRSemigroup", "NucleusOfFRSemigroup", "FRGroupPreImageData", "KneadingSequence", "Alphabet" ] gap> KnownPropertiesOfObject(BasilicaGroup); [ "IsDuplicateFree", "IsAssociative", "IsSimpleSemigroup", "IsFinitelyGeneratedGroup", "IsStateClosed", "IsBoundedFRSemigroup", "IsAmenableGroup" ] gap> KneadingSequence(BasilicaGroup); [/ '1', '*' ]