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): 144 Minimal number of generators: 25 Number of equivalence classes of cusps: 16 Genus: 5 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/1 5/4 3/2 5/3 2/1 5/2 8/3 30/11 3/1 10/3 4/1 5/1 6/1 20/3 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -6/1 -1/1 0/1 -5/1 -1/2 1/0 -4/1 -1/1 0/1 -3/1 -1/2 -14/5 -1/1 0/1 -11/4 1/0 -8/3 -1/1 -1/2 -5/2 -1/2 -2/1 -1/2 0/1 -5/3 -1/2 -8/5 -1/2 -1/3 -19/12 -3/8 -30/19 -1/3 -11/7 -1/4 -3/2 -1/2 -10/7 -1/3 -7/5 -1/4 -4/3 -1/3 0/1 -5/4 -1/2 -1/4 -6/5 -1/3 0/1 -13/11 -1/4 -20/17 0/1 -7/6 -1/2 -1/1 -1/4 0/1 0/1 1/1 1/4 6/5 0/1 1/3 5/4 1/4 1/2 4/3 0/1 1/3 3/2 1/2 14/9 0/1 1/3 11/7 1/4 8/5 1/3 1/2 5/3 1/2 2/1 0/1 1/2 5/2 1/2 8/3 1/2 1/1 19/7 3/4 30/11 1/1 11/4 1/0 3/1 1/2 10/3 1/1 7/2 1/0 4/1 0/1 1/1 5/1 1/2 1/0 6/1 0/1 1/1 13/2 1/0 20/3 0/1 7/1 1/2 1/0 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(11,80,-4,-29) (-6/1,1/0) -> (-14/5,-11/4) Hyperbolic Matrix(11,60,2,11) (-6/1,-5/1) -> (5/1,6/1) Hyperbolic Matrix(9,40,2,9) (-5/1,-4/1) -> (4/1,5/1) Hyperbolic Matrix(11,40,-8,-29) (-4/1,-3/1) -> (-7/5,-4/3) Hyperbolic Matrix(71,200,-60,-169) (-3/1,-14/5) -> (-6/5,-13/11) Hyperbolic Matrix(89,240,-56,-151) (-11/4,-8/3) -> (-8/5,-19/12) Hyperbolic Matrix(31,80,12,31) (-8/3,-5/2) -> (5/2,8/3) Hyperbolic Matrix(9,20,4,9) (-5/2,-2/1) -> (2/1,5/2) Hyperbolic Matrix(11,20,6,11) (-2/1,-5/3) -> (5/3,2/1) Hyperbolic Matrix(49,80,30,49) (-5/3,-8/5) -> (8/5,5/3) Hyperbolic Matrix(569,900,208,329) (-19/12,-30/19) -> (30/11,11/4) Hyperbolic Matrix(571,900,210,331) (-30/19,-11/7) -> (19/7,30/11) Hyperbolic Matrix(51,80,-44,-69) (-11/7,-3/2) -> (-7/6,-1/1) Hyperbolic Matrix(69,100,20,29) (-3/2,-10/7) -> (10/3,7/2) Hyperbolic Matrix(71,100,22,31) (-10/7,-7/5) -> (3/1,10/3) Hyperbolic Matrix(31,40,24,31) (-4/3,-5/4) -> (5/4,4/3) Hyperbolic Matrix(49,60,40,49) (-5/4,-6/5) -> (6/5,5/4) Hyperbolic Matrix(339,400,50,59) (-13/11,-20/17) -> (20/3,7/1) Hyperbolic Matrix(341,400,52,61) (-20/17,-7/6) -> (13/2,20/3) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(69,-80,44,-51) (1/1,6/5) -> (14/9,11/7) Hyperbolic Matrix(29,-40,8,-11) (4/3,3/2) -> (7/2,4/1) Hyperbolic Matrix(129,-200,20,-31) (3/2,14/9) -> (6/1,13/2) Hyperbolic Matrix(151,-240,56,-89) (11/7,8/5) -> (8/3,19/7) Hyperbolic Matrix(29,-80,4,-11) (11/4,3/1) -> (7/1,1/0) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(11,80,-4,-29) -> Matrix(1,0,0,1) Matrix(11,60,2,11) -> Matrix(1,0,2,1) Matrix(9,40,2,9) -> Matrix(1,0,2,1) Matrix(11,40,-8,-29) -> Matrix(1,0,-2,1) Matrix(71,200,-60,-169) -> Matrix(1,0,-2,1) Matrix(89,240,-56,-151) -> Matrix(3,2,-8,-5) Matrix(31,80,12,31) -> Matrix(3,2,4,3) Matrix(9,20,4,9) -> Matrix(1,0,4,1) Matrix(11,20,6,11) -> Matrix(1,0,4,1) Matrix(49,80,30,49) -> Matrix(5,2,12,5) Matrix(569,900,208,329) -> Matrix(11,4,8,3) Matrix(571,900,210,331) -> Matrix(13,4,16,5) Matrix(51,80,-44,-69) -> Matrix(1,0,0,1) Matrix(69,100,20,29) -> Matrix(5,2,2,1) Matrix(71,100,22,31) -> Matrix(7,2,10,3) Matrix(31,40,24,31) -> Matrix(1,0,6,1) Matrix(49,60,40,49) -> Matrix(1,0,6,1) Matrix(339,400,50,59) -> Matrix(1,0,6,1) Matrix(341,400,52,61) -> Matrix(1,0,2,1) Matrix(1,0,2,1) -> Matrix(1,0,8,1) Matrix(69,-80,44,-51) -> Matrix(1,0,0,1) Matrix(29,-40,8,-11) -> Matrix(1,0,-2,1) Matrix(129,-200,20,-31) -> Matrix(1,0,-2,1) Matrix(151,-240,56,-89) -> Matrix(5,-2,8,-3) Matrix(29,-80,4,-11) -> Matrix(1,0,0,1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 6 Minimal number of generators: 2 Number of equivalence classes of cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 6 Degree of the the map X: 6 Degree of the the map Y: 24 Permutation triple for Y: ((2,6,4,3,7)(5,13,10,9,8)(11,20,12,16,17)(14,22,18,19,21); (1,4,13,21,20,23,16,14,5,2)(3,10,15,8,7,17,22,24,19,11)(6,12)(9,18); (1,2,8,18,17,23,20,19,9,3)(4,12,21,24,22,16,6,5,15,10)(7,11)(13,14)) ----------------------------------------------------------------------- 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 The subgroup of modular group liftables which arise from translations is trivial. ----------------------------------------------------------------------- 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): 36 Minimal number of generators: 7 Number of equivalence classes of elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 8 Genus: 0 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 2/1 5/2 3/1 10/3 4/1 5/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES 0/1 0/1 1/1 1/4 5/4 1/4 1/2 4/3 0/1 1/3 3/2 1/2 5/3 1/2 2/1 0/1 1/2 5/2 1/2 3/1 1/2 10/3 1/1 7/2 1/0 4/1 0/1 1/1 5/1 1/2 1/0 1/0 1/0 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(9,-10,1,-1) (1/1,5/4) -> (5/1,1/0) Hyperbolic Matrix(31,-40,7,-9) (5/4,4/3) -> (4/1,5/1) Hyperbolic Matrix(29,-40,8,-11) (4/3,3/2) -> (7/2,4/1) Hyperbolic Matrix(19,-30,7,-11) (3/2,5/3) -> (5/2,3/1) Hyperbolic Matrix(11,-20,5,-9) (5/3,2/1) -> (2/1,5/2) Parabolic Matrix(31,-100,9,-29) (3/1,10/3) -> (10/3,7/2) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,0,1,1) -> Matrix(1,0,4,1) Matrix(9,-10,1,-1) -> Matrix(3,-1,4,-1) Matrix(31,-40,7,-9) -> Matrix(1,0,-2,1) Matrix(29,-40,8,-11) -> Matrix(1,0,-2,1) Matrix(19,-30,7,-11) -> Matrix(3,-1,4,-1) Matrix(11,-20,5,-9) -> Matrix(1,0,0,1) Matrix(31,-100,9,-29) -> Matrix(3,-2,2,-1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 3 Minimal number of generators: 2 Number of equivalence classes of elliptic points of order 2: 1 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 2 Genus: 0 Degree of H/liftables -> H/(image of liftables): 3 ----------------------------------------------------------------------- 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 This is a reflection group. CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE c d 0/1 0/1 4 1 2/1 (0/1,1/2) 0 5 5/2 1/2 1 2 3/1 1/2 1 10 10/3 1/1 2 1 4/1 (0/1,1/1) 0 5 5/1 (0/1,1/1).(1/2,1/0) 0 2 1/0 1/0 1 10 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(9,-20,4,-9) (2/1,5/2) -> (2/1,5/2) Reflection Matrix(11,-30,4,-11) (5/2,3/1) -> (5/2,3/1) Reflection Matrix(19,-60,6,-19) (3/1,10/3) -> (3/1,10/3) Reflection Matrix(11,-40,3,-11) (10/3,4/1) -> (10/3,4/1) Reflection Matrix(9,-40,2,-9) (4/1,5/1) -> (4/1,5/1) Reflection Matrix(-1,10,0,1) (5/1,1/0) -> (5/1,1/0) Reflection IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,0,0,-1) -> Matrix(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,1,-1) -> Matrix(1,0,4,-1) (0/1,2/1) -> (0/1,1/2) Matrix(9,-20,4,-9) -> Matrix(1,0,4,-1) (2/1,5/2) -> (0/1,1/2) Matrix(11,-30,4,-11) -> Matrix(-1,1,0,1) (5/2,3/1) -> (1/2,1/0) Matrix(19,-60,6,-19) -> Matrix(3,-2,4,-3) (3/1,10/3) -> (1/2,1/1) Matrix(11,-40,3,-11) -> Matrix(1,0,2,-1) (10/3,4/1) -> (0/1,1/1) Matrix(9,-40,2,-9) -> Matrix(1,0,2,-1) (4/1,5/1) -> (0/1,1/1) Matrix(-1,10,0,1) -> Matrix(-1,1,0,1) (5/1,1/0) -> (1/2,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.