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 1/2 5/8 3/4 5/6 1/1 5/4 3/2 5/3 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 -5/1 -1/1 -4/1 -2/3 -3/1 -1/2 -5/2 0/1 -2/1 0/1 -5/3 0/1 -3/2 1/1 -10/7 1/0 -7/5 1/0 -4/3 2/1 -5/4 1/0 -1/1 1/0 -5/6 -2/1 -4/5 -2/1 -3/4 -2/1 1/0 -5/7 -2/1 -2/3 -2/1 -5/8 -3/2 -3/5 -3/2 -10/17 -3/2 -7/12 -3/2 -4/3 -4/7 -4/3 -5/9 -1/1 -1/2 -1/1 0/1 -1/1 1/2 -1/1 4/7 -4/5 3/5 -3/4 5/8 -3/4 2/3 -2/3 3/4 -2/3 -1/2 7/9 -1/2 4/5 -2/3 5/6 -2/3 1/1 -1/2 5/4 -1/2 4/3 -2/5 3/2 -1/3 5/3 0/1 7/4 0/1 1/0 2/1 0/1 5/2 0/1 3/1 1/0 13/4 0/1 1/0 10/3 1/0 7/2 -1/1 4/1 -2/1 9/2 -1/1 5/1 -1/1 1/0 -1/1 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,10,0,1) (-5/1,1/0) -> (5/1,1/0) Parabolic Matrix(9,40,-16,-71) (-5/1,-4/1) -> (-4/7,-5/9) Hyperbolic Matrix(11,40,-8,-29) (-4/1,-3/1) -> (-7/5,-4/3) Hyperbolic Matrix(11,30,4,11) (-3/1,-5/2) -> (5/2,3/1) Hyperbolic Matrix(9,20,4,9) (-5/2,-2/1) -> (2/1,5/2) Hyperbolic Matrix(11,20,-16,-29) (-2/1,-5/3) -> (-5/7,-2/3) Hyperbolic Matrix(19,30,12,19) (-5/3,-3/2) -> (3/2,5/3) Hyperbolic Matrix(69,100,20,29) (-3/2,-10/7) -> (10/3,7/2) Hyperbolic Matrix(71,100,-120,-169) (-10/7,-7/5) -> (-3/5,-10/17) Hyperbolic Matrix(31,40,24,31) (-4/3,-5/4) -> (5/4,4/3) Hyperbolic Matrix(9,10,8,9) (-5/4,-1/1) -> (1/1,5/4) Hyperbolic Matrix(11,10,12,11) (-1/1,-5/6) -> (5/6,1/1) Hyperbolic Matrix(49,40,60,49) (-5/6,-4/5) -> (4/5,5/6) Hyperbolic Matrix(51,40,-88,-69) (-4/5,-3/4) -> (-7/12,-4/7) Hyperbolic Matrix(69,50,40,29) (-3/4,-5/7) -> (5/3,7/4) Hyperbolic Matrix(31,20,48,31) (-2/3,-5/8) -> (5/8,2/3) Hyperbolic Matrix(49,30,80,49) (-5/8,-3/5) -> (3/5,5/8) Hyperbolic Matrix(341,200,104,61) (-10/17,-7/12) -> (13/4,10/3) Hyperbolic Matrix(91,50,20,11) (-5/9,-1/2) -> (9/2,5/1) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(71,-40,16,-9) (1/2,4/7) -> (4/1,9/2) Hyperbolic Matrix(69,-40,88,-51) (4/7,3/5) -> (7/9,4/5) Hyperbolic Matrix(29,-20,16,-11) (2/3,3/4) -> (7/4,2/1) Hyperbolic Matrix(129,-100,40,-31) (3/4,7/9) -> (3/1,13/4) Hyperbolic Matrix(29,-40,8,-11) (4/3,3/2) -> (7/2,4/1) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,10,0,1) -> Matrix(1,0,0,1) Matrix(9,40,-16,-71) -> Matrix(7,6,-6,-5) Matrix(11,40,-8,-29) -> Matrix(1,0,2,1) Matrix(11,30,4,11) -> Matrix(1,0,2,1) Matrix(9,20,4,9) -> Matrix(1,0,0,1) Matrix(11,20,-16,-29) -> Matrix(3,2,-2,-1) Matrix(19,30,12,19) -> Matrix(1,0,-4,1) Matrix(69,100,20,29) -> Matrix(1,-2,0,1) Matrix(71,100,-120,-169) -> Matrix(3,-4,-2,3) Matrix(31,40,24,31) -> Matrix(1,-4,-2,9) Matrix(9,10,8,9) -> Matrix(1,2,-2,-3) Matrix(11,10,12,11) -> Matrix(1,4,-2,-7) Matrix(49,40,60,49) -> Matrix(3,4,-4,-5) Matrix(51,40,-88,-69) -> Matrix(3,2,-2,-1) Matrix(69,50,40,29) -> Matrix(1,2,0,1) Matrix(31,20,48,31) -> Matrix(7,12,-10,-17) Matrix(49,30,80,49) -> Matrix(5,6,-6,-7) Matrix(341,200,104,61) -> Matrix(3,4,2,3) Matrix(91,50,20,11) -> Matrix(1,0,0,1) Matrix(1,0,4,1) -> Matrix(3,4,-4,-5) Matrix(71,-40,16,-9) -> Matrix(7,6,-6,-5) Matrix(69,-40,88,-51) -> Matrix(3,2,-2,-1) Matrix(29,-20,16,-11) -> Matrix(3,2,-2,-1) Matrix(129,-100,40,-31) -> Matrix(3,2,-2,-1) Matrix(29,-40,8,-11) -> Matrix(1,0,2,1) 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): 7 Degree of the the map X: 7 Degree of the the map Y: 24 Permutation triple for Y: ((1,2)(3,10,8,7,18,6,15,5,4,11)(9,16,14,13,19,17,22,21,20,12)(23,24); (1,5,16,9,3,2,8,22,17,6)(4,14,24,19,18,7,21,23,12,11)(10,15)(13,20); (1,3,12,13,4)(2,6,19,20,7)(5,10,9,23,14)(8,15,17,24,21)) ----------------------------------------------------------------------- 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. ----------------------------------------------------------------------- The image of the modular group liftables in PSL(2,Z) equals the image of the pure modular group liftables. ----------------------------------------------------------------------- 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 -1/1 2 2 1/2 -1/1 2 10 4/7 -4/5 1 10 3/5 -3/4 1 10 5/8 -3/4 2 2 2/3 -2/3 1 10 3/4 (-2/3,-1/2) 0 10 7/9 -1/2 1 10 4/5 -2/3 1 10 5/6 -2/3 2 2 1/1 -1/2 1 10 5/4 -1/2 6 2 4/3 -2/5 1 10 3/2 -1/3 2 10 5/3 0/1 3 2 7/4 (0/1,1/0) 0 10 2/1 0/1 1 10 5/2 0/1 2 2 3/1 1/0 1 10 13/4 (0/1,1/0) 0 10 10/3 1/0 1 2 7/2 -1/1 2 10 4/1 -2/1 1 10 9/2 -1/1 2 10 5/1 -1/1 3 2 1/0 (-1/1,0/1) 0 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(1,0,4,-1) (0/1,1/2) -> (0/1,1/2) Reflection Matrix(71,-40,16,-9) (1/2,4/7) -> (4/1,9/2) Hyperbolic Matrix(69,-40,88,-51) (4/7,3/5) -> (7/9,4/5) Hyperbolic Matrix(49,-30,80,-49) (3/5,5/8) -> (3/5,5/8) Reflection Matrix(31,-20,48,-31) (5/8,2/3) -> (5/8,2/3) Reflection Matrix(29,-20,16,-11) (2/3,3/4) -> (7/4,2/1) Hyperbolic Matrix(129,-100,40,-31) (3/4,7/9) -> (3/1,13/4) Hyperbolic Matrix(49,-40,60,-49) (4/5,5/6) -> (4/5,5/6) Reflection Matrix(11,-10,12,-11) (5/6,1/1) -> (5/6,1/1) Reflection Matrix(9,-10,8,-9) (1/1,5/4) -> (1/1,5/4) Reflection Matrix(31,-40,24,-31) (5/4,4/3) -> (5/4,4/3) Reflection Matrix(29,-40,8,-11) (4/3,3/2) -> (7/2,4/1) Hyperbolic Matrix(19,-30,12,-19) (3/2,5/3) -> (3/2,5/3) Reflection Matrix(41,-70,24,-41) (5/3,7/4) -> (5/3,7/4) 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(79,-260,24,-79) (13/4,10/3) -> (13/4,10/3) Reflection Matrix(41,-140,12,-41) (10/3,7/2) -> (10/3,7/2) Reflection Matrix(19,-90,4,-19) (9/2,5/1) -> (9/2,5/1) 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(-1,0,2,1) (0/1,1/0) -> (-1/1,0/1) Matrix(1,0,4,-1) -> Matrix(5,4,-6,-5) (0/1,1/2) -> (-1/1,-2/3) Matrix(71,-40,16,-9) -> Matrix(7,6,-6,-5) -1/1 Matrix(69,-40,88,-51) -> Matrix(3,2,-2,-1) -1/1 Matrix(49,-30,80,-49) -> Matrix(7,6,-8,-7) (3/5,5/8) -> (-1/1,-3/4) Matrix(31,-20,48,-31) -> Matrix(17,12,-24,-17) (5/8,2/3) -> (-3/4,-2/3) Matrix(29,-20,16,-11) -> Matrix(3,2,-2,-1) -1/1 Matrix(129,-100,40,-31) -> Matrix(3,2,-2,-1) -1/1 Matrix(49,-40,60,-49) -> Matrix(5,4,-6,-5) (4/5,5/6) -> (-1/1,-2/3) Matrix(11,-10,12,-11) -> Matrix(7,4,-12,-7) (5/6,1/1) -> (-2/3,-1/2) Matrix(9,-10,8,-9) -> Matrix(3,2,-4,-3) (1/1,5/4) -> (-1/1,-1/2) Matrix(31,-40,24,-31) -> Matrix(9,4,-20,-9) (5/4,4/3) -> (-1/2,-2/5) Matrix(29,-40,8,-11) -> Matrix(1,0,2,1) 0/1 Matrix(19,-30,12,-19) -> Matrix(-1,0,6,1) (3/2,5/3) -> (-1/3,0/1) Matrix(41,-70,24,-41) -> Matrix(1,0,0,-1) (5/3,7/4) -> (0/1,1/0) Matrix(9,-20,4,-9) -> Matrix(-1,0,2,1) (2/1,5/2) -> (-1/1,0/1) Matrix(11,-30,4,-11) -> Matrix(1,0,0,-1) (5/2,3/1) -> (0/1,1/0) Matrix(79,-260,24,-79) -> Matrix(1,0,0,-1) (13/4,10/3) -> (0/1,1/0) Matrix(41,-140,12,-41) -> Matrix(1,2,0,-1) (10/3,7/2) -> (-1/1,1/0) Matrix(19,-90,4,-19) -> Matrix(5,6,-4,-5) (9/2,5/1) -> (-3/2,-1/1) Matrix(-1,10,0,1) -> Matrix(-1,0,2,1) (5/1,1/0) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.