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): 96 Minimal number of generators: 17 Number of equivalence classes of cusps: 12 Genus: 3 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -4/1 -3/2 0/1 1/1 3/2 12/7 2/1 3/1 7/2 4/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -4/1 -1/2 1/0 -7/2 -1/1 1/0 -3/1 -1/2 -2/1 -1/2 -9/5 -1/2 -7/4 -3/8 -1/3 -12/7 -1/3 -5/3 -3/10 -3/2 -1/4 -7/5 -1/6 -11/8 -1/4 -1/5 -4/3 -1/4 -1/6 -5/4 -1/4 0/1 -6/5 -1/4 -7/6 -1/4 -1/5 -1/1 -1/6 0/1 0/1 1/1 1/6 4/3 1/6 1/4 3/2 1/4 8/5 1/4 3/10 5/3 3/10 12/7 1/3 7/4 1/3 3/8 2/1 1/2 3/1 1/2 10/3 1/2 7/2 1/1 1/0 11/3 1/2 4/1 1/2 1/0 5/1 1/2 11/2 1/1 1/0 6/1 1/0 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(5,24,-4,-19) (-4/1,1/0) -> (-4/3,-5/4) Hyperbolic Matrix(19,72,-14,-53) (-4/1,-7/2) -> (-11/8,-4/3) Hyperbolic Matrix(25,84,-14,-47) (-7/2,-3/1) -> (-9/5,-7/4) Hyperbolic Matrix(5,12,2,5) (-3/1,-2/1) -> (2/1,3/1) Hyperbolic Matrix(53,96,16,29) (-2/1,-9/5) -> (3/1,10/3) Hyperbolic Matrix(97,168,56,97) (-7/4,-12/7) -> (12/7,7/4) Hyperbolic Matrix(71,120,42,71) (-12/7,-5/3) -> (5/3,12/7) Hyperbolic Matrix(23,36,-16,-25) (-5/3,-3/2) -> (-3/2,-7/5) Parabolic Matrix(43,60,-38,-53) (-7/5,-11/8) -> (-7/6,-1/1) Hyperbolic Matrix(29,36,4,5) (-5/4,-6/5) -> (6/1,1/0) Hyperbolic Matrix(91,108,16,19) (-6/5,-7/6) -> (11/2,6/1) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(19,-24,4,-5) (1/1,4/3) -> (4/1,5/1) Hyperbolic Matrix(25,-36,16,-23) (4/3,3/2) -> (3/2,8/5) Parabolic Matrix(67,-108,18,-29) (8/5,5/3) -> (11/3,4/1) Hyperbolic Matrix(47,-84,14,-25) (7/4,2/1) -> (10/3,7/2) Hyperbolic Matrix(43,-156,8,-29) (7/2,11/3) -> (5/1,11/2) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(5,24,-4,-19) -> Matrix(1,0,-4,1) Matrix(19,72,-14,-53) -> Matrix(1,0,-4,1) Matrix(25,84,-14,-47) -> Matrix(3,2,-8,-5) Matrix(5,12,2,5) -> Matrix(1,0,4,1) Matrix(53,96,16,29) -> Matrix(1,0,4,1) Matrix(97,168,56,97) -> Matrix(17,6,48,17) Matrix(71,120,42,71) -> Matrix(19,6,60,19) Matrix(23,36,-16,-25) -> Matrix(7,2,-32,-9) Matrix(43,60,-38,-53) -> Matrix(1,0,0,1) Matrix(29,36,4,5) -> Matrix(1,0,4,1) Matrix(91,108,16,19) -> Matrix(9,2,4,1) Matrix(1,0,2,1) -> Matrix(1,0,12,1) Matrix(19,-24,4,-5) -> Matrix(1,0,-4,1) Matrix(25,-36,16,-23) -> Matrix(9,-2,32,-7) Matrix(67,-108,18,-29) -> Matrix(7,-2,4,-1) Matrix(47,-84,14,-25) -> Matrix(5,-2,8,-3) Matrix(43,-156,8,-29) -> Matrix(1,0,0,1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 12 Minimal number of generators: 3 Number of equivalence classes of cusps: 4 Genus: 0 Degree of H/liftables -> H/(image of liftables): 3 Degree of the the map X: 6 Degree of the the map Y: 16 Permutation triple for Y: ((2,6,15,10,4,3)(5,7,13)(8,11,9)(12,16); (1,4,8,16,11,10,14,6,13,12,5,2)(3,9,15,7); (1,2,7,12,8,3)(4,11)(5,6)(9,16,13,15,14,10)) ----------------------------------------------------------------------- Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift elements of DeckMod(f) via pi_1: 0, lambda1 DeckMod(f) is isomorphic to Z/2Z. Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift modular group liftables via pi_1: 0, lambda1 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): 48 Minimal number of generators: 9 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: 1 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES 0/1 1/1 3/2 2/1 3/1 4/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -4/1 -1/2 1/0 -3/1 -1/2 -2/1 -1/2 -3/2 -1/4 -4/3 -1/4 -1/6 -5/4 -1/4 0/1 -6/5 -1/4 -1/1 -1/6 0/1 0/1 1/1 1/6 4/3 1/6 1/4 3/2 1/4 2/1 1/2 3/1 1/2 4/1 1/2 1/0 5/1 1/2 6/1 1/0 1/0 0/1 1/0 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(5,24,-4,-19) (-4/1,1/0) -> (-4/3,-5/4) Hyperbolic Matrix(7,24,2,7) (-4/1,-3/1) -> (3/1,4/1) Hyperbolic Matrix(5,12,2,5) (-3/1,-2/1) -> (2/1,3/1) Hyperbolic Matrix(7,12,4,7) (-2/1,-3/2) -> (3/2,2/1) Hyperbolic Matrix(17,24,12,17) (-3/2,-4/3) -> (4/3,3/2) Hyperbolic Matrix(29,36,4,5) (-5/4,-6/5) -> (6/1,1/0) Hyperbolic Matrix(31,36,6,7) (-6/5,-1/1) -> (5/1,6/1) Hyperbolic Matrix(1,0,2,1) (-1/1,0/1) -> (0/1,1/1) Parabolic Matrix(19,-24,4,-5) (1/1,4/3) -> (4/1,5/1) Hyperbolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(5,24,-4,-19) -> Matrix(1,0,-4,1) Matrix(7,24,2,7) -> Matrix(1,1,0,1) Matrix(5,12,2,5) -> Matrix(1,0,4,1) Matrix(7,12,4,7) -> Matrix(3,1,8,3) Matrix(17,24,12,17) -> Matrix(5,1,24,5) Matrix(29,36,4,5) -> Matrix(1,0,4,1) Matrix(31,36,6,7) -> Matrix(5,1,4,1) Matrix(1,0,2,1) -> Matrix(1,0,12,1) Matrix(19,-24,4,-5) -> Matrix(1,0,-4,1) 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 elliptic points of order 2: 0 Number of equivalence classes of elliptic points of order 3: 0 Number of equivalence classes of cusps: 3 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 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE c d 0/1 0/1 6 1 1/1 1/6 1 12 4/3 (1/6,1/4) 0 3 3/2 1/4 2 4 2/1 1/2 1 6 3/1 1/2 1 4 4/1 (1/2,1/0) 0 3 5/1 1/2 1 12 6/1 1/0 1 2 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(1,0,2,-1) (0/1,1/1) -> (0/1,1/1) Reflection Matrix(19,-24,4,-5) (1/1,4/3) -> (4/1,5/1) Hyperbolic Matrix(17,-24,12,-17) (4/3,3/2) -> (4/3,3/2) Reflection Matrix(7,-12,4,-7) (3/2,2/1) -> (3/2,2/1) Reflection Matrix(5,-12,2,-5) (2/1,3/1) -> (2/1,3/1) Reflection Matrix(7,-24,2,-7) (3/1,4/1) -> (3/1,4/1) Reflection Matrix(11,-60,2,-11) (5/1,6/1) -> (5/1,6/1) Reflection Matrix(-1,12,0,1) (6/1,1/0) -> (6/1,1/0) 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(1,0,12,-1) (0/1,1/1) -> (0/1,1/6) Matrix(19,-24,4,-5) -> Matrix(1,0,-4,1) 0/1 Matrix(17,-24,12,-17) -> Matrix(5,-1,24,-5) (4/3,3/2) -> (1/6,1/4) Matrix(7,-12,4,-7) -> Matrix(3,-1,8,-3) (3/2,2/1) -> (1/4,1/2) Matrix(5,-12,2,-5) -> Matrix(1,0,4,-1) (2/1,3/1) -> (0/1,1/2) Matrix(7,-24,2,-7) -> Matrix(-1,1,0,1) (3/1,4/1) -> (1/2,1/0) Matrix(11,-60,2,-11) -> Matrix(-1,1,0,1) (5/1,6/1) -> (1/2,1/0) Matrix(-1,12,0,1) -> Matrix(1,0,0,-1) (6/1,1/0) -> (0/1,1/0) ----------------------------------------------------------------------- The pullback map has no extra symmetries.