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): 288 Minimal number of generators: 49 Number of equivalence classes of cusps: 30 Genus: 10 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -4/7 -1/2 -8/21 -2/7 -15/56 -1/4 -2/11 -1/6 0/1 1/7 1/6 1/5 3/14 2/9 1/4 2/7 4/13 1/3 5/14 2/5 3/7 1/2 4/7 9/14 2/3 5/7 11/14 6/7 1/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 1/1 -4/5 0/1 1/0 -11/14 0/1 -7/9 0/1 1/1 -3/4 0/1 1/1 1/0 -11/15 0/1 1/1 -8/11 0/1 1/0 -5/7 0/1 -2/3 0/1 1/1 -9/14 1/1 -7/11 1/1 1/0 -12/19 0/1 1/0 -5/8 0/1 1/1 1/0 -13/21 0/1 -8/13 0/1 1/1 -11/18 0/1 1/1 1/0 -3/5 0/1 1/1 -4/7 1/1 -5/9 1/1 2/1 -6/11 0/1 1/0 -7/13 0/1 1/1 -1/2 0/1 1/1 1/0 -5/11 1/1 1/0 -4/9 2/1 1/0 -3/7 1/0 -2/5 0/1 1/0 -5/13 0/1 1/1 -8/21 1/1 -11/29 1/1 2/1 -3/8 1/1 2/1 1/0 -4/11 4/1 1/0 -5/14 1/0 -1/3 -1/1 1/0 -2/7 -1/1 1/1 -3/11 -1/1 1/0 -7/26 -2/1 -1/1 1/0 -11/41 -2/1 -1/1 -15/56 -1/1 -4/15 -1/1 0/1 -1/4 -1/1 0/1 1/0 -3/13 -1/1 0/1 -5/22 -1/1 -1/2 0/1 -2/9 -1/2 0/1 -3/14 0/1 -1/5 0/1 1/1 -2/11 2/1 1/0 -1/6 -1/1 0/1 1/0 -1/7 0/1 0/1 0/1 1/0 1/7 0/1 1/6 0/1 1/1 1/0 1/5 -1/1 0/1 3/14 0/1 2/9 0/1 1/2 3/13 0/1 1/1 1/4 0/1 1/1 1/0 2/7 -1/1 1/1 3/10 0/1 1/1 1/0 4/13 0/1 1/1 1/3 1/1 1/0 5/14 1/0 4/11 -4/1 1/0 7/19 -3/1 -2/1 3/8 -2/1 -1/1 1/0 2/5 0/1 1/0 3/7 1/0 4/9 -2/1 1/0 5/11 -1/1 1/0 6/13 -2/1 -1/1 1/2 -1/1 0/1 1/0 4/7 -1/1 7/12 -1/1 -2/3 -1/2 10/17 -2/3 -1/2 3/5 -1/1 0/1 8/13 -1/1 0/1 13/21 0/1 5/8 -1/1 0/1 1/0 17/27 -1/1 0/1 12/19 0/1 1/0 7/11 -1/1 1/0 9/14 -1/1 2/3 -1/1 0/1 5/7 0/1 8/11 0/1 1/0 19/26 -1/1 0/1 1/0 30/41 -2/1 -1/1 41/56 -1/1 11/15 -1/1 0/1 3/4 -1/1 0/1 1/0 10/13 -1/1 0/1 7/9 -1/1 0/1 11/14 0/1 4/5 0/1 1/0 5/6 -2/1 -1/1 1/0 6/7 -1/1 7/8 -1/1 -1/2 0/1 1/1 -1/1 0/1 1/0 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,1) (-1/1,1/0) -> (1/1,1/0) Parabolic Matrix(27,22,-70,-57) (-1/1,-4/5) -> (-2/5,-5/13) Hyperbolic Matrix(111,88,140,111) (-4/5,-11/14) -> (11/14,4/5) Hyperbolic Matrix(197,154,252,197) (-11/14,-7/9) -> (7/9,11/14) Hyperbolic Matrix(111,86,-182,-141) (-7/9,-3/4) -> (-11/18,-3/5) Hyperbolic Matrix(27,20,112,83) (-3/4,-11/15) -> (3/13,1/4) Hyperbolic Matrix(167,122,-308,-225) (-11/15,-8/11) -> (-6/11,-7/13) Hyperbolic Matrix(111,80,154,111) (-8/11,-5/7) -> (5/7,8/11) Hyperbolic Matrix(29,20,42,29) (-5/7,-2/3) -> (2/3,5/7) Hyperbolic Matrix(55,36,84,55) (-2/3,-9/14) -> (9/14,2/3) Hyperbolic Matrix(197,126,308,197) (-9/14,-7/11) -> (7/11,9/14) Hyperbolic Matrix(139,88,308,195) (-7/11,-12/19) -> (4/9,5/11) Hyperbolic Matrix(111,70,-490,-309) (-12/19,-5/8) -> (-5/22,-2/9) Hyperbolic Matrix(29,18,182,113) (-5/8,-13/21) -> (1/7,1/6) Hyperbolic Matrix(337,208,546,337) (-13/21,-8/13) -> (8/13,13/21) Hyperbolic Matrix(111,68,364,223) (-8/13,-11/18) -> (3/10,4/13) Hyperbolic Matrix(55,32,-98,-57) (-3/5,-4/7) -> (-4/7,-5/9) Parabolic Matrix(113,62,308,169) (-5/9,-6/11) -> (4/11,7/19) Hyperbolic Matrix(113,60,-420,-223) (-7/13,-1/2) -> (-7/26,-11/41) Hyperbolic Matrix(83,38,-308,-141) (-1/2,-5/11) -> (-3/11,-7/26) Hyperbolic Matrix(195,88,308,139) (-5/11,-4/9) -> (12/19,7/11) Hyperbolic Matrix(55,24,126,55) (-4/9,-3/7) -> (3/7,4/9) Hyperbolic Matrix(29,12,70,29) (-3/7,-2/5) -> (2/5,3/7) Hyperbolic Matrix(335,128,-882,-337) (-5/13,-8/21) -> (-8/21,-11/29) Parabolic Matrix(281,106,448,169) (-11/29,-3/8) -> (5/8,17/27) Hyperbolic Matrix(27,10,-154,-57) (-3/8,-4/11) -> (-2/11,-1/6) Hyperbolic Matrix(111,40,308,111) (-4/11,-5/14) -> (5/14,4/11) Hyperbolic Matrix(29,10,84,29) (-5/14,-1/3) -> (1/3,5/14) Hyperbolic Matrix(27,8,-98,-29) (-1/3,-2/7) -> (-2/7,-3/11) Parabolic Matrix(2297,616,3136,841) (-11/41,-15/56) -> (41/56,11/15) Hyperbolic Matrix(2295,614,3136,839) (-15/56,-4/15) -> (30/41,41/56) Hyperbolic Matrix(85,22,112,29) (-4/15,-1/4) -> (3/4,10/13) Hyperbolic Matrix(83,20,112,27) (-1/4,-3/13) -> (11/15,3/4) Hyperbolic Matrix(113,26,126,29) (-3/13,-5/22) -> (7/8,1/1) Hyperbolic Matrix(55,12,252,55) (-2/9,-3/14) -> (3/14,2/9) Hyperbolic Matrix(29,6,140,29) (-3/14,-1/5) -> (1/5,3/14) Hyperbolic Matrix(83,16,140,27) (-1/5,-2/11) -> (10/17,3/5) Hyperbolic Matrix(113,18,182,29) (-1/6,-1/7) -> (13/21,5/8) Hyperbolic Matrix(1,0,14,1) (-1/7,0/1) -> (0/1,1/7) Parabolic Matrix(57,-10,154,-27) (1/6,1/5) -> (7/19,3/8) Hyperbolic Matrix(309,-70,490,-111) (2/9,3/13) -> (17/27,12/19) Hyperbolic Matrix(29,-8,98,-27) (1/4,2/7) -> (2/7,3/10) Parabolic Matrix(83,-26,182,-57) (4/13,1/3) -> (5/11,6/13) Hyperbolic Matrix(57,-22,70,-27) (3/8,2/5) -> (4/5,5/6) Hyperbolic Matrix(307,-144,420,-197) (6/13,1/2) -> (19/26,30/41) Hyperbolic Matrix(57,-32,98,-55) (1/2,4/7) -> (4/7,7/12) Parabolic Matrix(419,-246,574,-337) (7/12,10/17) -> (8/11,19/26) Hyperbolic Matrix(141,-86,182,-111) (3/5,8/13) -> (10/13,7/9) Hyperbolic Matrix(85,-72,98,-83) (5/6,6/7) -> (6/7,7/8) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,0,1) -> Matrix(1,0,-2,1) Matrix(27,22,-70,-57) -> Matrix(1,0,0,1) Matrix(111,88,140,111) -> Matrix(1,0,0,1) Matrix(197,154,252,197) -> Matrix(1,0,-2,1) Matrix(111,86,-182,-141) -> Matrix(1,0,0,1) Matrix(27,20,112,83) -> Matrix(1,0,0,1) Matrix(167,122,-308,-225) -> Matrix(1,0,0,1) Matrix(111,80,154,111) -> Matrix(1,0,0,1) Matrix(29,20,42,29) -> Matrix(1,0,-2,1) Matrix(55,36,84,55) -> Matrix(1,0,-2,1) Matrix(197,126,308,197) -> Matrix(1,-2,0,1) Matrix(139,88,308,195) -> Matrix(1,-2,0,1) Matrix(111,70,-490,-309) -> Matrix(1,0,-2,1) Matrix(29,18,182,113) -> Matrix(1,0,0,1) Matrix(337,208,546,337) -> Matrix(1,0,-2,1) Matrix(111,68,364,223) -> Matrix(1,0,0,1) Matrix(55,32,-98,-57) -> Matrix(3,-2,2,-1) Matrix(113,62,308,169) -> Matrix(1,-4,0,1) Matrix(113,60,-420,-223) -> Matrix(1,-2,0,1) Matrix(83,38,-308,-141) -> Matrix(1,-2,0,1) Matrix(195,88,308,139) -> Matrix(1,-2,0,1) Matrix(55,24,126,55) -> Matrix(1,-4,0,1) Matrix(29,12,70,29) -> Matrix(1,0,0,1) Matrix(335,128,-882,-337) -> Matrix(3,-2,2,-1) Matrix(281,106,448,169) -> Matrix(1,-2,0,1) Matrix(27,10,-154,-57) -> Matrix(1,-2,0,1) Matrix(111,40,308,111) -> Matrix(1,-8,0,1) Matrix(29,10,84,29) -> Matrix(1,2,0,1) Matrix(27,8,-98,-29) -> Matrix(1,0,0,1) Matrix(2297,616,3136,841) -> Matrix(1,2,-2,-3) Matrix(2295,614,3136,839) -> Matrix(3,2,-2,-1) Matrix(85,22,112,29) -> Matrix(1,0,0,1) Matrix(83,20,112,27) -> Matrix(1,0,0,1) Matrix(113,26,126,29) -> Matrix(1,0,0,1) Matrix(55,12,252,55) -> Matrix(1,0,4,1) Matrix(29,6,140,29) -> Matrix(1,0,-2,1) Matrix(83,16,140,27) -> Matrix(1,0,-2,1) Matrix(113,18,182,29) -> Matrix(1,0,0,1) Matrix(1,0,14,1) -> Matrix(1,0,0,1) Matrix(57,-10,154,-27) -> Matrix(1,-2,0,1) Matrix(309,-70,490,-111) -> Matrix(1,0,-2,1) Matrix(29,-8,98,-27) -> Matrix(1,0,0,1) Matrix(83,-26,182,-57) -> Matrix(1,-2,0,1) Matrix(57,-22,70,-27) -> Matrix(1,0,0,1) Matrix(307,-144,420,-197) -> Matrix(1,0,0,1) Matrix(57,-32,98,-55) -> Matrix(1,2,-2,-3) Matrix(419,-246,574,-337) -> Matrix(3,2,-2,-1) Matrix(141,-86,182,-111) -> Matrix(1,0,0,1) Matrix(85,-72,98,-83) -> 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): 7 Degree of the the map X: 7 Degree of the the map Y: 48 Permutation triple for Y: ((1,4,16,42,17,5,2)(3,10,8,7,25,32,11)(6,21,27,29,9,14,13)(18,41,40,33,22,36,43)(20,38,37,46,26,31,35)(24,39,47,48,44,30,34); (1,2,8,21,45,40,46,48,47,31,43,28,9,3)(4,14,38,20,6,5,19,44,33,32,25,36,39,15)(7,23,37,16,41,18,17,35,12,11,30,29,27,24)(10,26)(13,22)(34,42); (2,6,22,44,46,23,7)(3,12,35,47,36,13,4)(5,18,31,10,9,30,19)(8,26,40,16,15,39,27)(11,33,45,21,20,17,34)(14,28,43,25,24,42,37)) ----------------------------------------------------------------------- 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 (0/1,1/0) 0 14 1/7 0/1 1 2 1/6 0 7 1/5 (-1/1,0/1) 0 14 3/14 0/1 3 1 2/9 (0/1,1/2) 0 14 3/13 (0/1,1/1) 0 14 1/4 0 7 2/7 0 2 3/10 0 7 4/13 (0/1,1/1) 0 14 1/3 (1/1,1/0) 0 14 5/14 1/0 5 1 4/11 (-4/1,1/0) 0 14 7/19 (-3/1,-2/1) 0 14 3/8 0 7 2/5 (0/1,1/0) 0 14 3/7 1/0 2 2 4/9 (-2/1,1/0) 0 14 5/11 (-1/1,1/0) 0 14 6/13 (-2/1,-1/1) 0 14 1/2 0 7 4/7 -1/1 2 2 7/12 0 7 10/17 (-2/3,-1/2) 0 14 3/5 (-1/1,0/1) 0 14 8/13 (-1/1,0/1) 0 14 13/21 0/1 1 2 5/8 0 7 17/27 (-1/1,0/1) 0 14 12/19 (0/1,1/0) 0 14 7/11 (-1/1,1/0) 0 14 9/14 -1/1 1 1 2/3 (-1/1,0/1) 0 14 5/7 0/1 1 2 8/11 (0/1,1/0) 0 14 19/26 0 7 30/41 (-2/1,-1/1) 0 14 41/56 -1/1 2 1 11/15 (-1/1,0/1) 0 14 3/4 0 7 10/13 (-1/1,0/1) 0 14 7/9 (-1/1,0/1) 0 14 11/14 0/1 1 1 4/5 (0/1,1/0) 0 14 5/6 0 7 6/7 -1/1 2 2 7/8 0 7 1/1 (-1/1,0/1) 0 14 1/0 0/1 1 1 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(1,0,14,-1) (0/1,1/7) -> (0/1,1/7) Reflection Matrix(113,-18,182,-29) (1/7,1/6) -> (13/21,5/8) Glide Reflection Matrix(57,-10,154,-27) (1/6,1/5) -> (7/19,3/8) Hyperbolic Matrix(29,-6,140,-29) (1/5,3/14) -> (1/5,3/14) Reflection Matrix(55,-12,252,-55) (3/14,2/9) -> (3/14,2/9) Reflection Matrix(309,-70,490,-111) (2/9,3/13) -> (17/27,12/19) Hyperbolic Matrix(83,-20,112,-27) (3/13,1/4) -> (11/15,3/4) Glide Reflection Matrix(29,-8,98,-27) (1/4,2/7) -> (2/7,3/10) Parabolic Matrix(139,-42,182,-55) (3/10,4/13) -> (3/4,10/13) Glide Reflection Matrix(83,-26,182,-57) (4/13,1/3) -> (5/11,6/13) Hyperbolic Matrix(29,-10,84,-29) (1/3,5/14) -> (1/3,5/14) Reflection Matrix(111,-40,308,-111) (5/14,4/11) -> (5/14,4/11) Reflection Matrix(223,-82,378,-139) (4/11,7/19) -> (10/17,3/5) Glide Reflection Matrix(57,-22,70,-27) (3/8,2/5) -> (4/5,5/6) Hyperbolic Matrix(29,-12,70,-29) (2/5,3/7) -> (2/5,3/7) Reflection Matrix(55,-24,126,-55) (3/7,4/9) -> (3/7,4/9) Reflection Matrix(195,-88,308,-139) (4/9,5/11) -> (12/19,7/11) Glide Reflection Matrix(307,-144,420,-197) (6/13,1/2) -> (19/26,30/41) Hyperbolic Matrix(57,-32,98,-55) (1/2,4/7) -> (4/7,7/12) Parabolic Matrix(419,-246,574,-337) (7/12,10/17) -> (8/11,19/26) Hyperbolic Matrix(141,-86,182,-111) (3/5,8/13) -> (10/13,7/9) Hyperbolic Matrix(337,-208,546,-337) (8/13,13/21) -> (8/13,13/21) Reflection Matrix(197,-124,224,-141) (5/8,17/27) -> (7/8,1/1) Glide Reflection Matrix(197,-126,308,-197) (7/11,9/14) -> (7/11,9/14) Reflection Matrix(55,-36,84,-55) (9/14,2/3) -> (9/14,2/3) Reflection Matrix(29,-20,42,-29) (2/3,5/7) -> (2/3,5/7) Reflection Matrix(111,-80,154,-111) (5/7,8/11) -> (5/7,8/11) Reflection Matrix(3361,-2460,4592,-3361) (30/41,41/56) -> (30/41,41/56) Reflection Matrix(1231,-902,1680,-1231) (41/56,11/15) -> (41/56,11/15) Reflection Matrix(197,-154,252,-197) (7/9,11/14) -> (7/9,11/14) Reflection Matrix(111,-88,140,-111) (11/14,4/5) -> (11/14,4/5) Reflection Matrix(85,-72,98,-83) (5/6,6/7) -> (6/7,7/8) Parabolic Matrix(-1,2,0,1) (1/1,1/0) -> (1/1,1/0) 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(1,0,14,-1) -> Matrix(1,0,0,-1) (0/1,1/7) -> (0/1,1/0) Matrix(113,-18,182,-29) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(57,-10,154,-27) -> Matrix(1,-2,0,1) 1/0 Matrix(29,-6,140,-29) -> Matrix(-1,0,2,1) (1/5,3/14) -> (-1/1,0/1) Matrix(55,-12,252,-55) -> Matrix(1,0,4,-1) (3/14,2/9) -> (0/1,1/2) Matrix(309,-70,490,-111) -> Matrix(1,0,-2,1) 0/1 Matrix(83,-20,112,-27) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(29,-8,98,-27) -> Matrix(1,0,0,1) Matrix(139,-42,182,-55) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(83,-26,182,-57) -> Matrix(1,-2,0,1) 1/0 Matrix(29,-10,84,-29) -> Matrix(-1,2,0,1) (1/3,5/14) -> (1/1,1/0) Matrix(111,-40,308,-111) -> Matrix(1,8,0,-1) (5/14,4/11) -> (-4/1,1/0) Matrix(223,-82,378,-139) -> Matrix(1,2,-2,-5) Matrix(57,-22,70,-27) -> Matrix(1,0,0,1) Matrix(29,-12,70,-29) -> Matrix(1,0,0,-1) (2/5,3/7) -> (0/1,1/0) Matrix(55,-24,126,-55) -> Matrix(1,4,0,-1) (3/7,4/9) -> (-2/1,1/0) Matrix(195,-88,308,-139) -> Matrix(1,2,0,-1) *** -> (-1/1,1/0) Matrix(307,-144,420,-197) -> Matrix(1,0,0,1) Matrix(57,-32,98,-55) -> Matrix(1,2,-2,-3) -1/1 Matrix(419,-246,574,-337) -> Matrix(3,2,-2,-1) -1/1 Matrix(141,-86,182,-111) -> Matrix(1,0,0,1) Matrix(337,-208,546,-337) -> Matrix(-1,0,2,1) (8/13,13/21) -> (-1/1,0/1) Matrix(197,-124,224,-141) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(197,-126,308,-197) -> Matrix(1,2,0,-1) (7/11,9/14) -> (-1/1,1/0) Matrix(55,-36,84,-55) -> Matrix(-1,0,2,1) (9/14,2/3) -> (-1/1,0/1) Matrix(29,-20,42,-29) -> Matrix(-1,0,2,1) (2/3,5/7) -> (-1/1,0/1) Matrix(111,-80,154,-111) -> Matrix(1,0,0,-1) (5/7,8/11) -> (0/1,1/0) Matrix(3361,-2460,4592,-3361) -> Matrix(3,4,-2,-3) (30/41,41/56) -> (-2/1,-1/1) Matrix(1231,-902,1680,-1231) -> Matrix(-1,0,2,1) (41/56,11/15) -> (-1/1,0/1) Matrix(197,-154,252,-197) -> Matrix(-1,0,2,1) (7/9,11/14) -> (-1/1,0/1) Matrix(111,-88,140,-111) -> Matrix(1,0,0,-1) (11/14,4/5) -> (0/1,1/0) Matrix(85,-72,98,-83) -> Matrix(1,2,-2,-3) -1/1 Matrix(-1,2,0,1) -> Matrix(-1,0,2,1) (1/1,1/0) -> (-1/1,0/1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.