INFORMATION ON THE IMAGE IN PSL(2,Z) OF THE GROUP OF PURE MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE Index in PSL(2,Z): 48 Minimal number of generators: 9 Number of equivalence classes of cusps: 6 Genus: 2 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/1 2/1 7/3 7/2 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -3/1 3/14 -2/1 2/7 -7/4 1/3 -5/3 5/14 -3/2 3/7 -7/5 1/2 -4/3 4/7 -1/1 1/0 0/1 0/1 1/1 1/14 3/2 3/35 2/1 2/21 7/3 1/10 5/2 5/49 3/1 3/28 7/2 1/9 4/1 4/35 1/0 1/7 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(3,14,-2,-9) (-3/1,1/0) -> (-5/3,-3/2) Hyperbolic Matrix(5,14,-4,-11) (-3/1,-2/1) -> (-4/3,-1/1) Hyperbolic Matrix(23,42,6,11) (-2/1,-7/4) -> (7/2,4/1) Hyperbolic Matrix(33,56,10,17) (-7/4,-5/3) -> (3/1,7/2) Hyperbolic Matrix(39,56,16,23) (-3/2,-7/5) -> (7/3,5/2) Hyperbolic Matrix(31,42,14,19) (-7/5,-4/3) -> (2/1,7/3) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(11,-14,4,-5) (1/1,3/2) -> (5/2,3/1) Hyperbolic Matrix(9,-14,2,-3) (3/2,2/1) -> (4/1,1/0) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(3,14,-2,-9) -> Matrix(11,-2,28,-5) Matrix(5,14,-4,-11) -> Matrix(9,-2,14,-3) Matrix(23,42,6,11) -> Matrix(19,-6,168,-53) Matrix(33,56,10,17) -> Matrix(23,-8,210,-73) Matrix(39,56,16,23) -> Matrix(17,-8,168,-79) Matrix(31,42,14,19) -> Matrix(11,-6,112,-61) Matrix(1,0,2,1) -> Matrix(1,0,14,1) Matrix(11,-14,4,-5) -> Matrix(25,-2,238,-19) Matrix(9,-14,2,-3) -> Matrix(23,-2,196,-17) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 48 Minimal number of generators: 9 Number of equivalence classes of cusps: 6 Genus: 2 Degree of H/liftables -> H/(image of liftables): 1 Degree of the the map X: 8 Degree of the the map Y: 8 Permutation triple for Y: ((2,6,7,4,3,8,5); (1,4,3,7,6,5,2); (1,2,6,5,8,4,3)) ----------------------------------------------------------------------- Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift elements of DeckMod(f) via pi_1: 0 DeckMod(f) is trivial. Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift modular group liftables via pi_1: 0, lambda1, lambda2, lambda1+lambda2 The subgroup of modular group liftables which arise from translations is isomorphic to Z/2Z+Z/2Z. ----------------------------------------------------------------------- INFORMATION ON THE IMAGE IN PSL(2,Z) OF THE GROUP OF MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE Index in PSL(2,Z): 8 Minimal number of generators: 3 Number of equivalence classes of elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 2 Number of equivalence classes of cusps: 2 Genus: 0 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 1/14 2/1 2/21 1/0 1/7 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,1,1) (0/1,1/0) -> (0/1,1/1) Parabolic Matrix(5,-7,3,-4) (1/1,5/3) -> (4/3,2/1) Elliptic Matrix(3,-7,1,-2) (2/1,4/1) -> (5/2,1/0) Elliptic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,7,1) Matrix(5,-7,3,-4) -> Matrix(12,-1,133,-11) Matrix(3,-7,1,-2) -> Matrix(10,-1,91,-9) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 8 Minimal number of generators: 3 Number of equivalence classes of elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 2 Number of equivalence classes of cusps: 2 Genus: 0 Degree of H/liftables -> H/(image of liftables): 1 ----------------------------------------------------------------------- INFORMATION ON THE IMAGE IN PGL(2,Z) OF THE GROUP OF EXTENDED MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE c d 0/1 0/1 7 1 2/1 2/21 1 7 1/0 1/7 1 7 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Reflection Matrix(1,0,1,-1) (0/1,2/1) -> (0/1,2/1) Reflection Matrix(3,-7,1,-2) (2/1,4/1) -> (5/2,1/0) Elliptic IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,14,-1) (0/1,1/0) -> (0/1,1/7) Matrix(1,0,1,-1) -> Matrix(1,0,21,-1) (0/1,2/1) -> (0/1,2/21) Matrix(3,-7,1,-2) -> Matrix(10,-1,91,-9) (1/10,1/8).(0/1,2/19).(1/11,1/9) ----------------------------------------------------------------------- The pullback map was not drawn because this NET map is Euclidean.