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: 20 Genus: 3 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -4/1 -5/2 -2/1 -20/11 -5/4 0/1 1/1 5/4 10/7 3/2 5/3 2/1 5/2 3/1 10/3 7/2 4/1 9/2 5/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -5/1 0/1 -9/2 0/1 1/1 -4/1 1/0 -11/3 1/0 -7/2 -1/1 0/1 -10/3 0/1 -3/1 1/0 -5/2 0/1 -7/3 1/2 -9/4 0/1 1/1 -2/1 1/0 -11/6 -2/1 -1/1 -20/11 -1/1 -9/5 1/0 -7/4 -2/1 -1/1 -5/3 -1/1 -3/2 -1/1 0/1 -10/7 -1/1 -7/5 -3/4 -11/8 -1/1 -2/3 -4/3 -1/2 -5/4 -2/3 0/1 -6/5 -1/2 -1/1 -1/2 0/1 0/1 1/1 1/2 5/4 0/1 2/3 9/7 1/2 4/3 1/2 11/8 2/3 1/1 7/5 3/4 10/7 1/1 3/2 0/1 1/1 5/3 1/1 7/4 1/1 2/1 9/5 1/0 2/1 1/0 5/2 0/1 8/3 1/2 19/7 1/2 30/11 1/1 11/4 0/1 1/1 3/1 1/0 10/3 0/1 7/2 0/1 1/1 11/3 1/0 4/1 1/0 9/2 -1/1 0/1 5/1 0/1 1/0 0/1 1/0 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(19,90,4,19) (-5/1,-9/2) -> (9/2,5/1) Hyperbolic Matrix(19,80,14,59) (-9/2,-4/1) -> (4/3,11/8) Hyperbolic Matrix(21,80,16,61) (-4/1,-11/3) -> (9/7,4/3) Hyperbolic Matrix(39,140,22,79) (-11/3,-7/2) -> (7/4,9/5) Hyperbolic Matrix(41,140,12,41) (-7/2,-10/3) -> (10/3,7/2) Hyperbolic Matrix(19,60,6,19) (-10/3,-3/1) -> (3/1,10/3) Hyperbolic Matrix(19,50,-8,-21) (-3/1,-5/2) -> (-5/2,-7/3) Parabolic Matrix(61,140,44,101) (-7/3,-9/4) -> (11/8,7/5) Hyperbolic Matrix(19,40,-10,-21) (-9/4,-2/1) -> (-2/1,-11/6) Parabolic Matrix(301,550,110,201) (-11/6,-20/11) -> (30/11,11/4) Hyperbolic Matrix(359,650,132,239) (-20/11,-9/5) -> (19/7,30/11) Hyperbolic Matrix(79,140,22,39) (-9/5,-7/4) -> (7/2,11/3) Hyperbolic Matrix(41,70,24,41) (-7/4,-5/3) -> (5/3,7/4) Hyperbolic Matrix(19,30,12,19) (-5/3,-3/2) -> (3/2,5/3) Hyperbolic Matrix(41,60,28,41) (-3/2,-10/7) -> (10/7,3/2) Hyperbolic Matrix(99,140,70,99) (-10/7,-7/5) -> (7/5,10/7) Hyperbolic Matrix(79,110,28,39) (-7/5,-11/8) -> (11/4,3/1) Hyperbolic Matrix(59,80,14,19) (-11/8,-4/3) -> (4/1,9/2) Hyperbolic Matrix(39,50,-32,-41) (-4/3,-5/4) -> (-5/4,-6/5) Parabolic Matrix(59,70,16,19) (-6/5,-1/1) -> (11/3,4/1) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(41,-50,32,-39) (1/1,5/4) -> (5/4,9/7) Parabolic Matrix(59,-110,22,-41) (9/5,2/1) -> (8/3,19/7) Hyperbolic Matrix(21,-50,8,-19) (2/1,5/2) -> (5/2,8/3) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,10,0,1) -> Matrix(1,0,0,1) Matrix(19,90,4,19) -> Matrix(1,0,-2,1) Matrix(19,80,14,59) -> Matrix(1,-2,2,-3) Matrix(21,80,16,61) -> Matrix(1,0,2,1) Matrix(39,140,22,79) -> Matrix(1,2,0,1) Matrix(41,140,12,41) -> Matrix(1,0,2,1) Matrix(19,60,6,19) -> Matrix(1,0,0,1) Matrix(19,50,-8,-21) -> Matrix(1,0,2,1) Matrix(61,140,44,101) -> Matrix(1,-2,2,-3) Matrix(19,40,-10,-21) -> Matrix(1,-2,0,1) Matrix(301,550,110,201) -> Matrix(1,2,0,1) Matrix(359,650,132,239) -> Matrix(1,0,2,1) Matrix(79,140,22,39) -> Matrix(1,2,0,1) Matrix(41,70,24,41) -> Matrix(3,4,2,3) Matrix(19,30,12,19) -> Matrix(1,0,2,1) Matrix(41,60,28,41) -> Matrix(1,0,2,1) Matrix(99,140,70,99) -> Matrix(7,6,8,7) Matrix(79,110,28,39) -> Matrix(3,2,4,3) Matrix(59,80,14,19) -> Matrix(3,2,-2,-1) Matrix(39,50,-32,-41) -> Matrix(1,0,0,1) Matrix(59,70,16,19) -> Matrix(1,0,2,1) Matrix(1,0,2,1) -> Matrix(1,0,4,1) Matrix(41,-50,32,-39) -> Matrix(1,0,0,1) Matrix(59,-110,22,-41) -> Matrix(1,-2,2,-3) Matrix(21,-50,8,-19) -> 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): 6 Degree of the the map X: 6 Degree of the the map Y: 24 Permutation triple for Y: ((2,6,18,19,7)(3,11,23,12,4)(5,15,10,9,16)(8,21,14,13,22); (1,4,14,21,23,24,18,15,5,2)(3,10,20,8,7,19,13,17,16,11)(6,12)(9,22); (1,2,8,9,3)(4,6,5,17,13)(10,18,12,21,20)(16,22,19,24,23)) ----------------------------------------------------------------------- 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 0/1 2 1 1/1 1/2 1 10 5/4 0 2 9/7 1/2 1 10 4/3 1/2 1 5 11/8 (2/3,1/1) 0 10 7/5 3/4 1 10 10/7 1/1 3 1 3/2 (0/1,1/1) 0 10 5/3 1/1 2 2 7/4 (1/1,2/1) 0 10 9/5 1/0 1 10 2/1 1/0 1 5 5/2 0/1 2 2 8/3 1/2 1 5 19/7 1/2 1 10 30/11 1/1 1 1 11/4 (0/1,1/1) 0 10 3/1 1/0 1 10 10/3 0/1 1 1 7/2 (0/1,1/1) 0 10 11/3 1/0 1 10 4/1 1/0 1 5 9/2 (-1/1,0/1) 0 10 5/1 0/1 1 2 1/0 (0/1,1/0) 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(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(41,-50,32,-39) (1/1,5/4) -> (5/4,9/7) Parabolic Matrix(61,-80,16,-21) (9/7,4/3) -> (11/3,4/1) Glide Reflection Matrix(59,-80,14,-19) (4/3,11/8) -> (4/1,9/2) Glide Reflection Matrix(79,-110,28,-39) (11/8,7/5) -> (11/4,3/1) Glide Reflection Matrix(99,-140,70,-99) (7/5,10/7) -> (7/5,10/7) Reflection Matrix(41,-60,28,-41) (10/7,3/2) -> (10/7,3/2) Reflection 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(79,-140,22,-39) (7/4,9/5) -> (7/2,11/3) Glide Reflection Matrix(59,-110,22,-41) (9/5,2/1) -> (8/3,19/7) Hyperbolic Matrix(21,-50,8,-19) (2/1,5/2) -> (5/2,8/3) Parabolic Matrix(419,-1140,154,-419) (19/7,30/11) -> (19/7,30/11) Reflection Matrix(241,-660,88,-241) (30/11,11/4) -> (30/11,11/4) Reflection Matrix(19,-60,6,-19) (3/1,10/3) -> (3/1,10/3) 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(1,0,0,-1) (0/1,1/0) -> (0/1,1/0) Matrix(1,0,2,-1) -> Matrix(1,0,4,-1) (0/1,1/1) -> (0/1,1/2) Matrix(41,-50,32,-39) -> Matrix(1,0,0,1) Matrix(61,-80,16,-21) -> Matrix(1,0,2,-1) *** -> (0/1,1/1) Matrix(59,-80,14,-19) -> Matrix(3,-2,-2,1) Matrix(79,-110,28,-39) -> Matrix(3,-2,4,-3) *** -> (1/2,1/1) Matrix(99,-140,70,-99) -> Matrix(7,-6,8,-7) (7/5,10/7) -> (3/4,1/1) Matrix(41,-60,28,-41) -> Matrix(1,0,2,-1) (10/7,3/2) -> (0/1,1/1) Matrix(19,-30,12,-19) -> Matrix(1,0,2,-1) (3/2,5/3) -> (0/1,1/1) Matrix(41,-70,24,-41) -> Matrix(3,-4,2,-3) (5/3,7/4) -> (1/1,2/1) Matrix(79,-140,22,-39) -> Matrix(-1,2,0,1) *** -> (1/1,1/0) Matrix(59,-110,22,-41) -> Matrix(1,-2,2,-3) 1/1 Matrix(21,-50,8,-19) -> Matrix(1,0,2,1) 0/1 Matrix(419,-1140,154,-419) -> Matrix(3,-2,4,-3) (19/7,30/11) -> (1/2,1/1) Matrix(241,-660,88,-241) -> Matrix(1,0,2,-1) (30/11,11/4) -> (0/1,1/1) Matrix(19,-60,6,-19) -> Matrix(1,0,0,-1) (3/1,10/3) -> (0/1,1/0) Matrix(41,-140,12,-41) -> Matrix(1,0,2,-1) (10/3,7/2) -> (0/1,1/1) Matrix(19,-90,4,-19) -> Matrix(-1,0,2,1) (9/2,5/1) -> (-1/1,0/1) Matrix(-1,10,0,1) -> Matrix(1,0,0,-1) (5/1,1/0) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.