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): 192 Minimal number of generators: 33 Number of equivalence classes of cusps: 24 Genus: 5 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -9/4 -2/1 -3/2 -1/1 -3/4 -2/3 0/1 1/2 6/11 3/5 2/3 3/4 1/1 6/5 5/4 3/2 12/7 7/4 2/1 9/4 5/2 11/4 3/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -3/1 0/1 -11/4 0/1 1/3 -8/3 1/2 -5/2 0/1 1/1 -12/5 0/1 -7/3 1/2 -9/4 1/1 -2/1 1/0 -7/4 -1/1 1/0 -12/7 -1/1 -5/3 1/0 -3/2 -1/1 -7/5 -1/2 -18/13 -1/1 -11/8 -1/1 -2/3 -4/3 -1/2 -5/4 -1/2 0/1 -6/5 0/1 -1/1 -1/2 -3/4 -1/3 -5/7 -1/4 -12/17 -1/3 -7/10 -1/3 -2/7 -2/3 -1/4 -5/8 -1/4 0/1 -3/5 -1/4 -7/12 -1/4 -1/5 -4/7 -1/4 -9/16 -1/5 -5/9 -1/4 -6/11 -1/5 -1/2 -1/5 0/1 0/1 0/1 1/2 0/1 1/5 6/11 1/5 5/9 1/4 4/7 1/4 3/5 1/4 5/8 0/1 1/4 2/3 1/4 3/4 1/3 4/5 1/2 1/1 1/2 6/5 0/1 5/4 0/1 1/2 4/3 1/2 3/2 1/1 8/5 1/0 13/8 0/1 1/1 18/11 1/1 5/3 1/0 12/7 1/1 7/4 1/1 1/0 9/5 1/0 2/1 1/0 9/4 -1/1 16/7 -1/2 7/3 -1/2 12/5 0/1 5/2 -1/1 0/1 18/7 -1/1 13/5 -1/2 8/3 -1/2 11/4 -1/3 0/1 3/1 0/1 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,6,0,1) (-3/1,1/0) -> (3/1,1/0) Parabolic Matrix(23,66,8,23) (-3/1,-11/4) -> (11/4,3/1) Hyperbolic Matrix(71,192,44,119) (-11/4,-8/3) -> (8/5,13/8) Hyperbolic Matrix(25,66,-36,-95) (-8/3,-5/2) -> (-7/10,-2/3) Hyperbolic Matrix(49,120,20,49) (-5/2,-12/5) -> (12/5,5/2) Hyperbolic Matrix(23,54,20,47) (-12/5,-7/3) -> (1/1,6/5) Hyperbolic Matrix(47,108,-84,-193) (-7/3,-9/4) -> (-9/16,-5/9) Hyperbolic Matrix(25,54,-44,-95) (-9/4,-2/1) -> (-4/7,-9/16) Hyperbolic Matrix(23,42,-40,-73) (-2/1,-7/4) -> (-7/12,-4/7) Hyperbolic Matrix(97,168,56,97) (-7/4,-12/7) -> (12/7,7/4) Hyperbolic Matrix(71,120,-100,-169) (-12/7,-5/3) -> (-5/7,-12/17) Hyperbolic Matrix(23,36,-16,-25) (-5/3,-3/2) -> (-3/2,-7/5) Parabolic Matrix(95,132,172,239) (-7/5,-18/13) -> (6/11,5/9) Hyperbolic Matrix(313,432,192,265) (-18/13,-11/8) -> (13/8,18/11) Hyperbolic Matrix(97,132,36,49) (-11/8,-4/3) -> (8/3,11/4) Hyperbolic Matrix(23,30,36,47) (-4/3,-5/4) -> (5/8,2/3) Hyperbolic Matrix(49,60,40,49) (-5/4,-6/5) -> (6/5,5/4) Hyperbolic Matrix(47,54,20,23) (-6/5,-1/1) -> (7/3,12/5) Hyperbolic Matrix(23,18,-32,-25) (-1/1,-3/4) -> (-3/4,-5/7) Parabolic Matrix(265,186,104,73) (-12/17,-7/10) -> (5/2,18/7) Hyperbolic Matrix(47,30,36,23) (-2/3,-5/8) -> (5/4,4/3) Hyperbolic Matrix(49,30,80,49) (-5/8,-3/5) -> (3/5,5/8) Hyperbolic Matrix(143,84,80,47) (-3/5,-7/12) -> (7/4,9/5) Hyperbolic Matrix(217,120,132,73) (-5/9,-6/11) -> (18/11,5/3) Hyperbolic Matrix(23,12,44,23) (-6/11,-1/2) -> (1/2,6/11) Hyperbolic Matrix(1,0,4,1) (-1/2,0/1) -> (0/1,1/2) Parabolic Matrix(193,-108,84,-47) (5/9,4/7) -> (16/7,7/3) Hyperbolic Matrix(73,-42,40,-23) (4/7,3/5) -> (9/5,2/1) Hyperbolic Matrix(25,-18,32,-23) (2/3,3/4) -> (3/4,4/5) Parabolic Matrix(73,-60,28,-23) (4/5,1/1) -> (13/5,8/3) Hyperbolic Matrix(25,-36,16,-23) (4/3,3/2) -> (3/2,8/5) Parabolic Matrix(145,-246,56,-95) (5/3,12/7) -> (18/7,13/5) Hyperbolic Matrix(73,-162,32,-71) (2/1,9/4) -> (9/4,16/7) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,6,0,1) -> Matrix(1,0,0,1) Matrix(23,66,8,23) -> Matrix(1,0,-6,1) Matrix(71,192,44,119) -> Matrix(1,0,-2,1) Matrix(25,66,-36,-95) -> Matrix(3,-2,-10,7) Matrix(49,120,20,49) -> Matrix(1,0,-2,1) Matrix(23,54,20,47) -> Matrix(1,0,0,1) Matrix(47,108,-84,-193) -> Matrix(1,0,-6,1) Matrix(25,54,-44,-95) -> Matrix(1,-2,-4,9) Matrix(23,42,-40,-73) -> Matrix(1,0,-4,1) Matrix(97,168,56,97) -> Matrix(1,2,0,1) Matrix(71,120,-100,-169) -> Matrix(1,2,-4,-7) Matrix(23,36,-16,-25) -> Matrix(1,2,-2,-3) Matrix(95,132,172,239) -> Matrix(1,0,6,1) Matrix(313,432,192,265) -> Matrix(3,2,4,3) Matrix(97,132,36,49) -> Matrix(3,2,-8,-5) Matrix(23,30,36,47) -> Matrix(1,0,6,1) Matrix(49,60,40,49) -> Matrix(1,0,4,1) Matrix(47,54,20,23) -> Matrix(1,0,0,1) Matrix(23,18,-32,-25) -> Matrix(5,2,-18,-7) Matrix(265,186,104,73) -> Matrix(7,2,-4,-1) Matrix(47,30,36,23) -> Matrix(1,0,6,1) Matrix(49,30,80,49) -> Matrix(1,0,8,1) Matrix(143,84,80,47) -> Matrix(9,2,4,1) Matrix(217,120,132,73) -> Matrix(9,2,4,1) Matrix(23,12,44,23) -> Matrix(1,0,10,1) Matrix(1,0,4,1) -> Matrix(1,0,10,1) Matrix(193,-108,84,-47) -> Matrix(1,0,-6,1) Matrix(73,-42,40,-23) -> Matrix(1,0,-4,1) Matrix(25,-18,32,-23) -> Matrix(7,-2,18,-5) Matrix(73,-60,28,-23) -> Matrix(1,0,-4,1) Matrix(25,-36,16,-23) -> Matrix(3,-2,2,-1) Matrix(145,-246,56,-95) -> Matrix(1,0,-2,1) Matrix(73,-162,32,-71) -> Matrix(1,2,-2,-3) 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): 9 Degree of the the map X: 9 Degree of the the map Y: 32 Permutation triple for Y: ((1,2)(3,10,21,20,26,11)(4,13,30,19,14,5)(6,17,12,24,9,18)(7,16,15,28,22,8)(23,27)(25,29)(31,32); (1,5,16,27,26,20,31,15,14,29,17,6)(2,8,22,25,10,24,32,30,13,23,9,3)(4,11,28,12)(7,21,19,18); (1,3,4)(2,6,19,32,20,7)(5,15)(9,10)(11,27,13,12,29,22)(14,21,25)(16,18,23)(24,28,31)) ----------------------------------------------------------------------- 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 5 2 1/2 (0/1,1/5) 0 12 6/11 1/5 2 2 5/9 1/4 1 12 4/7 1/4 1 6 3/5 1/4 1 4 5/8 (0/1,1/4) 0 12 2/3 1/4 2 6 3/4 1/3 4 4 4/5 1/2 2 6 1/1 1/2 1 12 6/5 0/1 1 2 5/4 (0/1,1/2) 0 12 4/3 1/2 2 6 3/2 1/1 2 4 8/5 1/0 2 6 13/8 (0/1,1/1) 0 12 18/11 1/1 2 2 5/3 1/0 1 12 12/7 1/1 1 2 7/4 (1/1,1/0) 0 12 9/5 1/0 1 4 2/1 1/0 1 6 9/4 -1/1 4 4 16/7 -1/2 1 6 7/3 -1/2 1 12 12/5 0/1 1 2 5/2 (-1/1,0/1) 0 12 18/7 -1/1 1 2 13/5 -1/2 1 12 8/3 -1/2 2 6 11/4 (-1/3,0/1) 0 12 3/1 0/1 3 4 1/0 (0/1,1/0) 0 12 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(23,-12,44,-23) (1/2,6/11) -> (1/2,6/11) Reflection Matrix(217,-120,132,-73) (6/11,5/9) -> (18/11,5/3) Glide Reflection Matrix(193,-108,84,-47) (5/9,4/7) -> (16/7,7/3) Hyperbolic Matrix(73,-42,40,-23) (4/7,3/5) -> (9/5,2/1) Hyperbolic Matrix(49,-30,80,-49) (3/5,5/8) -> (3/5,5/8) Reflection Matrix(47,-30,36,-23) (5/8,2/3) -> (5/4,4/3) Glide Reflection Matrix(25,-18,32,-23) (2/3,3/4) -> (3/4,4/5) Parabolic Matrix(73,-60,28,-23) (4/5,1/1) -> (13/5,8/3) Hyperbolic Matrix(47,-54,20,-23) (1/1,6/5) -> (7/3,12/5) Glide Reflection Matrix(49,-60,40,-49) (6/5,5/4) -> (6/5,5/4) Reflection Matrix(25,-36,16,-23) (4/3,3/2) -> (3/2,8/5) Parabolic Matrix(119,-192,44,-71) (8/5,13/8) -> (8/3,11/4) Glide Reflection Matrix(287,-468,176,-287) (13/8,18/11) -> (13/8,18/11) Reflection Matrix(145,-246,56,-95) (5/3,12/7) -> (18/7,13/5) Hyperbolic Matrix(97,-168,56,-97) (12/7,7/4) -> (12/7,7/4) Reflection Matrix(71,-126,40,-71) (7/4,9/5) -> (7/4,9/5) Reflection Matrix(73,-162,32,-71) (2/1,9/4) -> (9/4,16/7) Parabolic Matrix(49,-120,20,-49) (12/5,5/2) -> (12/5,5/2) Reflection Matrix(71,-180,28,-71) (5/2,18/7) -> (5/2,18/7) Reflection Matrix(23,-66,8,-23) (11/4,3/1) -> (11/4,3/1) Reflection Matrix(-1,6,0,1) (3/1,1/0) -> (3/1,1/0) 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(1,0,4,-1) -> Matrix(1,0,10,-1) (0/1,1/2) -> (0/1,1/5) Matrix(23,-12,44,-23) -> Matrix(1,0,10,-1) (1/2,6/11) -> (0/1,1/5) Matrix(217,-120,132,-73) -> Matrix(9,-2,4,-1) Matrix(193,-108,84,-47) -> Matrix(1,0,-6,1) 0/1 Matrix(73,-42,40,-23) -> Matrix(1,0,-4,1) 0/1 Matrix(49,-30,80,-49) -> Matrix(1,0,8,-1) (3/5,5/8) -> (0/1,1/4) Matrix(47,-30,36,-23) -> Matrix(1,0,6,-1) *** -> (0/1,1/3) Matrix(25,-18,32,-23) -> Matrix(7,-2,18,-5) 1/3 Matrix(73,-60,28,-23) -> Matrix(1,0,-4,1) 0/1 Matrix(47,-54,20,-23) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(49,-60,40,-49) -> Matrix(1,0,4,-1) (6/5,5/4) -> (0/1,1/2) Matrix(25,-36,16,-23) -> Matrix(3,-2,2,-1) 1/1 Matrix(119,-192,44,-71) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(287,-468,176,-287) -> Matrix(1,0,2,-1) (13/8,18/11) -> (0/1,1/1) Matrix(145,-246,56,-95) -> Matrix(1,0,-2,1) 0/1 Matrix(97,-168,56,-97) -> Matrix(-1,2,0,1) (12/7,7/4) -> (1/1,1/0) Matrix(71,-126,40,-71) -> Matrix(-1,2,0,1) (7/4,9/5) -> (1/1,1/0) Matrix(73,-162,32,-71) -> Matrix(1,2,-2,-3) -1/1 Matrix(49,-120,20,-49) -> Matrix(-1,0,2,1) (12/5,5/2) -> (-1/1,0/1) Matrix(71,-180,28,-71) -> Matrix(-1,0,2,1) (5/2,18/7) -> (-1/1,0/1) Matrix(23,-66,8,-23) -> Matrix(-1,0,6,1) (11/4,3/1) -> (-1/3,0/1) Matrix(-1,6,0,1) -> Matrix(1,0,0,-1) (3/1,1/0) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.