Fractals/Mathematics/group/Basilica group

right|thumb|Basilica JUlia set and external rays right|thumb|Lamination of Basilica Julia set right|thumb|NucleusMachine(BasilicaGroup) Basilica group is :A Thompson Group for the Basilica by James Belk, Bradley Forrest group defined by automatum the iterated monodromy group of the polynomial R. I. Grigorchuk and A. Zuk (2002a). On a torsion-free weakly branch group defined by a three state automaton. Internat. J. Algebra Comput., 12(1-2):223–246. International Conference on Geometric and Combinatorial Methods in Group Theory and Semigroup Theory (Lincoln, NE, 2000). related with Basilica Julia set : "the scaling limit of the Schreier graphs of its action on level n of T is the basilica"Amenability via random walks Laurent Bartholdi and Balint Virag May 19, 2003 Computation The critical points of the polynomial are and . The postcritical set is 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); <state-closed group over [ 1, 2 ] with 2 generators> gap> AssignGeneratorVariables(B); #I Assigned the global variables [ "a", "b" ] gap> B=BasilicaGroup; #I \=: converting second argument to FR element #I \<: converting second argument to FR element #I \<: converting second argument to FR element #I \=: converting second argument to FR element #I \=: converting second argument to FR element #I \<: converting second argument to FR element #I \<: converting second argument to FR element #I \=: converting second argument to FR element #I \=: converting first argument to FR element #I \=: converting first argument to FR element #I \=: converting first argument to FR element #I \=: converting first argument to FR element #I \=: converting first argument to FR element #I \=: converting first argument to FR element #I \=: converting first argument to FR element #I \=: converting first argument to FR element true 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', '*' ] References