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69A119F96B9BF0677F3367CBB3CD2FF0687DC9FC46243B5B1640E8E992893A 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSL8 %!PS-AdobeFont-1.1: CMSL8 1.0 %%CreationDate: 1991 Aug 20 16:41:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSL8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -9.46 def /isFixedPitch false def end readonly def /FontName /CMSL8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-57 -250 1183 750}readonly def /UniqueID 5000796 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 55FBD18DCEF738EFB8E96B4B61D9C1A24F5FE2B29EF89EA4320BCB1D16E30E22 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /square put readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueID 5031981 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY5 %!PS-AdobeFont-1.1: CMSY5 1.0 %%CreationDate: 1991 Aug 15 07:21:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{21 -944 1448 791}readonly def /UniqueID 5000815 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{0 -250 1171 750}readonly def /UniqueID 5087382 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D77639DF1232A4D6233A9CAF69B151DFD33F C0962EAC6E3EBFB8AD256A3C654EAAF9A50C51BC6FA90B61B60401C235AFAB7B 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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EUFM10 2.1 %%CreationDate: 1992 Nov 20 17:36:20 % Euler fonts were designed by Hermann Zapf. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (EUFM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 110 /n put dup 111 /o put dup 115 /s put dup 117 /u put readonly def /FontBBox{-26 -224 1055 741}readonly def /UniqueID 5031986 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 3 /bracketrightbig put dup 10 /angbracketleftbig put dup 11 /angbracketrightbig put dup 12 /vextendsingle put dup 14 /slashbig put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 26 /braceleftbigg put dup 27 /bracerightbigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 56 /bracelefttp put dup 57 /bracerighttp put dup 58 /braceleftbt put dup 59 /bracerightbt put dup 60 /braceleftmid put dup 61 /bracerightmid put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 101 /tildewide put 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All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueID 5000817 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D31FF2B87F97C73D63EECDDA4C49501773468A 27D1663E0B62F461F6E40A5D6676D1D12B51E641C1D4E8E2771864FC104F8CBF 5B78EC1D88228725F1C453A678F58A7E1B7BD7CA700717D288EB8DA1F57C4F09 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 5250011D19E9366EB6FD153D3A100CAA6212E3D5D93990737F8D326D347B7EDC 4391C9DF440285B8FC159D0E98D4258FC57892DDF753642CD526A96ACEDA4120 788F22B1D09F149794E66DD1AC2C2B3BC6FEC59D626F427CD5AE9C54C7F78F62 C36F49B3C2E5E62AFB56DCEE87445A12A942C14AE618D1FE1B11A9CF9FAA1F32 617B598CE5058715EF3051E228F72F651040AD99A741F247C68007E68C84E9D1 D0BF99AA5D777D88A7D3CED2EA67F4AE61E8BC0495E7DA382E82DDB2B009DD63 532C74E3BE5EC555A014BCBB6AB31B8286D7712E0E926F8696830672B8214E9B 5D0740C16ADF0AFD47C4938F373575C6CA91E46D88DE24E682DEC44B57EA8AF8 4E57D45646073250D82C4B50CBBB0B369932618301F3D4186277103B53B3C9E6 DB42D6B30115F67B9D078220D5752644930643BDF9FACF684EBE13E39B65055E B1BD054C324962025EC79E1D155936FE32D9F2224353F2A46C3558EF216F6BB2 A304BAF752BEEC36C4440B556AEFECF454BA7CBBA7537BCB10EBC21047333A89 8936419D857CD9F59EBA20B0A3D9BA4A0D3395336B4CDA4BA6451B6E4D1370FA D9BDABB7F271BC1C6C48D9DF1E5A6FAE788F5609DE3C48D47A67097C547D9817 AD3A7CCE2B771843D69F860DA4059A71494281C0AD8D4BAB3F67BB6739723C04 AE05F9E35B2B2CB9C7874C114F57A185C8563C0DCCA93F8096384D71A2994748 A3C7C8B8AF54961A8838AD279441D9A5EB6C1FE26C98BD025F353124DA68A827 AE2AF8D25CA48031C242AA433EEEBB8ABA4B96821786C38BACB5F58C3D5DA011 85B385124A4E61AEB68F3178EAB6FB6A6F6902D21AA7C08584F7021CCDD4E176 B50DDB1AD630BFDEA76B03FF92416050A9F13EA77405EBFE1C981CC3D436A09B F6BBA850F39CA7D69D75BED4B330251289F4762C7EA7DA9AEF5E6ED813B06FA7 5413AE9E719206DCF5C58F0765821BB6539EC9260519F969047B3F84EFFE2E70 D3BD8EFD227AA337A8ACEA6294B8BDC853BAF09336202B70254A3B06C482D0AA D4C443DEA7C56FDA047E790A488AA114444CB6F86EAE1862983604C42D9398D4 34D8A27B5E1BD40295FCE954AC424125C6E71AF982D0B15656C61A3B2C5FBA7D 9BB4C7007611872140F0365D7E739BC1C3BFF71CD0969D0D01A7281EE5E5AA79 F32D0684228CAB06F2F8AA242A150764FF124A26BC1DBB3921B91C5D2386B423 8B756D1F1DB9FB13327D45C0CFE9AD6245EFFB19F53AA254EF89E4B5D2CC44A8 CB482DC351AF4EE44548147C6E3F1224B684D82CB5D6E92DD43C7EE8BABD0CDA AC972E2B83CBDFFF84C4B1B4EB65FBF93502B375D4B32CC10FD7F711C56F57BB 4D01E44F688D59138785B16D2B36580580EBCC1562BBE7AC66F5DE6F74BC1D93 0996454774578CA93761B15A7408C8C3F6DC966540E58CA18CD4922D9AD09C12 7B5674FC30D51B477EB860AE2CAE5290FCD137318035055107E941A3C67C7674 54A0C39B238969910AEBD0C24F1362C06061EE7B248D1DC056BBDA0407C9C61F 14D73CE46B1076C91FB88F3166CCF4660371827ACAF90B413E0F4EB3680C466D 465B8E4B572BBFB60ED54F3BCB29EB246637C8279B0DA4DDA985DDD80D68F783 A9E42CC2EA2014B069C59115E482A0E7F36918C9886F3C107165BC7D6579EE5D FB4C140870AEE6C0BF8CAD9608424DEB1695D95CB5095CFCD407526649782F9C C093121866886E38946C392ECCD2A0B96BFD3527A77E4819E5E8A7092A6617DB 4341AD27ED2752CA2369BF2A0D522B98C73CA3C2BC38169ADAE59C278CF325A5 9A932CAB84BB18E8E57A983284246D594A9275E687B4E010EB1DB6803FE03D46 519D1D06F1E2DAA389C69DF288E78BE7E6F739BD3998DA30F61BDA8051949E7C 72C24014B3542A09B95189721233C58FC25E6DFBA0AC157D397BF3CC1AE73ACF 3F7046BFA834C8D5DAA3376529A3273DBB0E2005C85B38400E7ACBB29E47ED27 D4375E81CD 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF4E9D2405B169CD5365D6ECED5D768D66D6C 68618B8C482B341F8CA38E9BB9BAFCFAAD9C2F3FD033B62690986ED43D9C9361 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Fi(\023)456 1144 y Fp(to)22 b(the)h(univ)n(ersal)f(co)n(v)n(ering.)33 b(Ob)n(viously)-7 b(,)22 b Fm( )k Fp(is)d(orien)n(tation)e(rev)n(ersing.)34 b(Moreo)n(v)n(er,)21 b(conjuga-)456 1250 y(tion)d(with)h Fm( )j Fp(maps)c(elemen)n(ts)g(of)1508 1228 y Fi(e)1497 1250 y Fp(Sl\(2)p Fm(;)c Fg(R)p Fp(\))19 b(to)1892 1228 y Fi(e)1880 1250 y Fp(Sl\(2)p Fm(;)14 b Fg(R)p Fp(\).)34 b(Hence)19 b Fm( )i Fp(is)e(really)e(an)i(automorphism)456 1361 y(of)31 b(the)h(Th)n(urston)f(geometry)g(\()1479 1339 y Fi(e)1467 1361 y Fp(Sl\(2)p Fm(;)14 b Fg(R)p Fp(\))p Fm(;)1788 1339 y Fi(e)1776 1361 y Fp(Sl\(2)p Fm(;)g Fg(R)p Fp(\)\).)50 b(The)32 b(con)n(tact)f(structures)g Fm( )3072 1373 y Fj(\003)3110 1361 y Fp(\()p Fk(C)3186 1373 y Ff(st)3247 1361 y Fp(\))h(and)456 1472 y Fk(C)500 1484 y Ff(st)584 1472 y Fp(induce)24 b(di\013eren)n(t)h(orien)n(tations.)34 b(Th)n(us)24 b(all)g(geometric)e(con)n(tact)i(structures)f(on)3146 1450 y Fi(e)3134 1472 y Fp(Sl\(2\))h(are)456 1583 y(equiv)-5 b(alen)n(t)27 b(to)g(subgeometries)g(of)g(\()1626 1561 y Fi(e)1614 1583 y Fp(Sl)q(\(2\))p Fm(;)14 b Fk(C)1871 1595 y Ff(st)1931 1583 y Fm(;)g(G)p Fp(\).)555 1694 y(In)20 b(order)e(to)i(pro)n(v)n(e)d(that)j(the)g(geometric)f(con)n(tact)f (structure)h Fk(C)2523 1706 y Ff(st)2603 1694 y Fp(on)2722 1672 y Fi(e)2711 1694 y Fp(Sl\(2)p Fm(;)14 b Fg(R)p Fp(\))19 b(is)h(equiv)-5 b(alen)n(t)456 1794 y(to)25 b(the)i(tigh)n(t)e(\(cf.)i ([Be]\))f(con)n(tact)f(structure)g Fk(C)1933 1763 y Fj(0)1979 1794 y Fp(=)e(k)n(er)o(\()p Fm(dz)2296 1763 y Fj(0)2334 1794 y Fk(\000)14 b Fm(x)2460 1763 y Fj(0)2498 1794 y Fm(dy)2585 1763 y Fj(0)2608 1794 y Fp(\))26 b(on)g Fg(R)2840 1763 y Fn(3)2903 1794 y Fp(w)n(e)f(\014rst)h(exhibit)456 1904 y(a)h(con)n(v)n(ex)f(surface)1083 1884 y Fi(e)1076 1904 y Fp(\006)i(in)1272 1883 y Fi(e)1261 1904 y Fp(Sl\(2)p Fm(;)14 b Fg(R)p Fp(\))27 b(suc)n(h)g(that)h(the)g(line)g(\014eld)g Fm(T)2473 1884 y Fi(e)2468 1904 y Fp(\006)17 b Fk(\\)h(C)2662 1916 y Ff(st)2750 1904 y Fp(is)27 b(equiv)-5 b(alen)n(t)28 b(to)f(the)456 2004 y(line)h(\014eld)h(on)f(\006)970 1974 y Fj(0)1017 2004 y Fp(=)c Fk(f)p Fm(x)1195 1974 y Fj(0)1243 2004 y Fm(>)g Fp(0)p Fm(;)14 b(z)1454 1974 y Fj(0)1500 2004 y Fp(=)24 b(0)p Fk(g)f(\032)h Fg(R)1845 1974 y Fn(3)1911 2004 y Fp(induced)29 b(b)n(y)f Fk(C)2386 1974 y Fj(0)2409 2004 y Fp(.)39 b(In)28 b(addition)h(w)n(e)f(\014nd)g (a)g(Reeb)456 2114 y(v)n(ector)21 b(\014eld)i Fm(R)h Fp(transv)n(erse)c(to)1454 2093 y Fi(e)1447 2114 y Fp(\006)j(whose)f (\015o)n(w)g(is)h(conjugate)f(to)h(the)g(\015o)n(w)f(of)h(the)g(Reeb)g (v)n(ector)456 2220 y(\014eld)k Fm(@)679 2232 y Ff(z)713 2215 y Fe(0)768 2220 y Fp(of)g Fm(\013)915 2189 y Fj(0)967 2220 y Fp(suc)n(h)g(that)h(eac)n(h)f(\015o)n(w)g(line)h(of)f Fm(R)i Fp(in)n(tersects)2411 2199 y Fi(e)2404 2220 y Fp(\006)f(exactly)f(once.)555 2319 y(No)n(w)i(consider)f(the)i(con)n (tact)f(v)n(ector)f(\014eld)h(induced)h(b)n(y)f(horizon)n(tal)e(lifts)j (of)g(rotations)e(of)456 2419 y Fg(H)521 2389 y Fn(2)589 2419 y Fp(around)i(a)h(p)r(oin)n(t)h(in)g(the)f(in)n(terior)g(of)g Fg(H)1876 2389 y Fn(2)1913 2419 y Fp(.)48 b(By)31 b(de\014nition)h(the) g(lifts)g(of)f(these)h(rotations)456 2518 y(preserv)n(e)25 b(the)j(con)n(tact)f(structure)g(and)h(the)g(corresp)r(onding)d(v)n (ector)i(\014eld)h(are)e(transv)n(erse)g(to)456 2624 y Fk(C)500 2636 y Ff(st)560 2624 y Fp(.)47 b(Eac)n(h)30 b(orbit)g(of)h(the)h(\015o)n(w)e(of)h Fm(R)h Fp(in)n(tersects)2038 2603 y Fi(e)2031 2624 y Fp(\006)f(exactly)f(once)h(\(note)g(that)g(the) h(rotation)456 2735 y(of)27 b Fg(H)615 2705 y Fn(2)680 2735 y Fp(b)n(y)g(2)p Fm(\031)j Fp(yields)e(a)f(dec)n(k)g (transformation)f(of)2078 2713 y Fi(e)2067 2735 y Fp(Sl\(2)p Fm(;)14 b Fg(R)p Fp(\)\).)555 2845 y(F)-7 b(or)706 2824 y Fi(e)699 2845 y Fp(\006)22 b(w)n(e)g(tak)n(e)f(the)i(lift)g(of)f(the) g(surface)f(\006)i(whose)e(p)r(oin)n(ts)h(in)g(Sl)q(\(2)p Fm(;)14 b Fg(R)p Fp(\))22 b(are)f(represen)n(ted)g(b)n(y)456 2944 y(the)k(sp)r(eed)h(v)n(ector)e(\014eld)i(of)f(geo)r(desics)f 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Fp(\006)c(this)g(determines)f(the) h(con)n(tact)f(structure)g(on)2107 3549 y Fi(e)2095 3570 y Fp(Sl\(2)p Fm(;)14 b Fg(R)p Fp(\))24 b(up)g(to)f(isotop)n(y)-7 b(.)35 b(Since)23 b(the)h(\015o)n(ws)456 3681 y(of)35 b Fm(@)602 3693 y Ff(z)636 3677 y Fe(0)698 3681 y Fp(and)g Fm(R)h Fp(are)f(conjugate)f(this)i(implies)f(that)h(the)g(con)n(tact)e (structure)h(on)3093 3659 y Fi(e)3081 3681 y Fp(Sl\(2)p Fm(;)14 b Fg(R)p Fp(\))36 b(is)456 3781 y(equiv)-5 b(alen)n(t)27 b(to)g(the)h(standard)f(tigh)n(t)h(con)n(tact)f(structure)g(on)g Fg(R)2466 3751 y Fn(3)2503 3781 y Fp(.)456 4012 y(2.4.)40 b Fm(X)g Fp(=)33 b(Sol)944 3974 y Fn(3)981 4012 y Ft(.)41 b Fp(The)34 b(group)e(Sol)1578 3974 y Fn(3)1649 4012 y Fp(is)i(the)g(semidirect)g(pro)r(duct)f Fg(R)2669 3981 y Fn(2)2729 4012 y Fg(n)22 b(R)p Fp(:)50 b(W)-7 b(e)34 b(write)f Fm(x;)14 b(y)456 4111 y Fp(for)27 b(the)h(co)r(ordinates)f (on)g Fg(R)1343 4081 y Fn(2)1409 4111 y Fp(and)g Fm(t)h Fp(for)g(the)g(co)r(ordinate)f(on)h Fg(R)p Fp(.)37 b(The)28 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2922 y Fn(3)2309 2953 y Fp(.)43 b(In)30 b(particular,)f(ev)n(ery)g(subgroup)456 3052 y Fm(H)h Fk(\032)23 b Fm(G)28 b Fp(acting)f(transitiv)n(ely)g(on)g Fg(R)1597 3022 y Fn(3)1662 3052 y Fp(m)n(ust)h(con)n(tain)f(the)i (translations)d Fg(R)2818 3022 y Fn(3)2855 3052 y Fp(,)i(cf.)38 b(Theorem)27 b(3.)456 3152 y(But)c(ev)n(ery)f(plane)h(\014eld)g(whic)n (h)g(is)g(in)n(v)-5 b(arian)n(t)22 b(under)h(translations)f(is)h(in)n (tegrable.)34 b(Th)n(us)23 b(there)456 3251 y(is)k(no)g(geometric)g (con)n(tact)g(structure)g(on)g Fg(R)1857 3221 y Fn(3)1894 3251 y Fp(.)456 3441 y(2.6.)40 b Fm(X)j Fp(=)37 b Fg(H)905 3411 y Fn(3)942 3441 y Ft(.)k Fp(The)35 b(metric)h(on)g Fm(X)43 b Fp(=)36 b Fg(H)1857 3411 y Fn(3)1930 3441 y Fp(has)f(constan)n(t)g(sectional)g(curv)-5 b(ature)35 b(and)h(the)456 3541 y(maximal)d(isometry)g(group)g(is)g Fm(G)h Fk(')f Fp(PSl\(2)p Fm(;)14 b Fg(C)p Fp(\))23 b Fg(n)f(Z)2169 3553 y Fn(2)2240 3541 y Fp(where)34 b(the)g(generator)e (of)i Fg(Z)3168 3553 y Fn(2)3239 3541 y Fp(is)g(the)456 3641 y(re\015ection)27 b(of)g Fg(H)975 3610 y Fn(3)1040 3641 y Fp(along)f(the)i(plane)f Fg(H)1690 3610 y Fn(2)1750 3641 y Fk(\032)c Fg(H)1903 3610 y Fn(3)1940 3641 y Fp(.)555 3740 y(By)29 b(Theorem)g(3,)g(\()p Fg(H)1231 3710 y Fn(3)1268 3740 y Fm(;)14 b Fp(PSL\(2)p Fm(;)g Fg(C)p Fp(\)\))29 b(do)r(es)g(not)h(ha)n(v)n(e)e(an)n(y)g(prop)r(er)h(subgeometries.)40 b(Since)456 3840 y(the)20 b(stabilizer)f(of)h(a)f(p)r(oin)n(t)h(in)g Fg(H)1450 3810 y Fn(3)1507 3840 y Fp(is)f(isomorphic)g(to)h(SO)o (\(3\),)i(there)e(is)f(no)h(plane)g(\014eld)g(in)n(v)-5 b(arian)n(t)456 3939 y(under)31 b(all)g(elemen)n(ts)g(of)g(PSL)o(\(2)p Fm(;)14 b Fg(R)p Fp(\).)48 b(Th)n(us)31 b(there)g(is)g(no)g(geometric)f (con)n(tact)g(structure)h(on)456 4039 y Fg(H)521 4009 y Fn(3)557 4039 y Fp(.)456 4229 y(2.7.)40 b Fm(X)33 b Fp(=)26 b Fm(S)875 4199 y Fn(2)931 4229 y Fk(\002)20 b Fg(R)p Ft(.)41 b Fp(This)29 b(is)h(the)g(ob)n(vious)e(pro)r(duct)i (geometry)-7 b(.)41 b(The)30 b(full)g(isometry)f(group)456 4328 y(is)e(the)h(pro)r(duct)g(of)f(the)h(isometry)f(groups)f(of)i Fm(S)1990 4298 y Fn(2)2054 4328 y Fp(and)g Fg(R)p Fp(.)555 4428 y(Supp)r(ose)e(that)f Fk(C)30 b Fp(is)25 b(a)g(geometric)f(con)n (tact)g(structure)h(on)g Fm(X)7 b Fp(.)36 b(Since)25 b Fm(G)h Fp(acts)e(transitiv)n(ely)-7 b(,)456 4528 y Fk(C)27 b Fp(is)22 b(either)g(ev)n(erywhere)f(tangen)n(t)h(to)g(the)h (\014b)r(ers)f(of)g(the)h(pro)5 b(jection)22 b Fm(S)2685 4498 y Fn(2)2730 4528 y Fk(\002)8 b Fg(R)22 b Fp(on)n(to)f(the)i (second)456 4627 y(factor)28 b(or)h(ev)n(erywhere)f(transv)n(erse)f(to) j(the)g(\014b)r(ers.)42 b(On)30 b(the)g(one)f(hand)g(con)n(tact)g (structures)456 4727 y(ha)n(v)n(e)d(no)h(in)n(tegral)f(surfaces)g(and)h (on)g(the)h(other)f(hand)g Fm(S)2275 4697 y Fn(2)2340 4727 y Fp(do)r(es)g(not)g(admit)h(a)e(non{singular)456 4827 y(line)h(\014eld.)38 b(Th)n(us)27 b(there)g(is)h(no)f(geometric)g (con)n(tact)f(structure)i(on)f Fm(S)2645 4796 y Fn(2)2700 4827 y Fk(\002)18 b Fm(R)q Fp(.)456 5016 y(2.8.)40 b Fm(X)30 b Fp(=)22 b Fg(H)877 4986 y Fn(2)921 5016 y Fk(\002)7 b Fg(R)p Ft(.)40 b Fp(Again,)23 b(this)f(is)g(the)g(ob)n(vious)e(pro)r (duct)i(geometry)-7 b(.)34 b(The)22 b(isometry)f(group)456 5116 y Fm(G)27 b Fp(is)g(the)h(pro)r(duct)f(of)g(the)g(isometry)f (groups)g(of)h(the)h(factors.)35 b(The)27 b(dimension)g(of)g Fm(G)h Fp(is)f(four)456 5216 y(and)g Fm(G)682 5228 y Fn(0)743 5216 y Fp(=)22 b(PSl\(2)p Fm(;)14 b Fg(R)p Fp(\))k Fk(\002)g Fg(R)p Fp(.)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 456 251 a Fn(10)1266 b(T.)23 b(V)n(OGEL)555 450 y Fp(By)31 b(Theorem)e(3,)j(this)f(geometry)e(has)h(no)g(prop)r(er) g(subgeometry)g(of)g(dimension)h(3.)45 b(The)456 550 y(only)31 b(plane)g(\014eld)h(whic)n(h)f(is)h(in)n(v)-5 b(arian)n(t)30 b(under)i(the)f(action)g(of)h(PSl\(2)p Fm(;)14 b Fg(R)p Fp(\))31 b(is)h(tangen)n(t)f(to)g(the)456 649 y(\014rst)20 b(factor)g(of)g Fg(H)1003 619 y Fn(2)1044 649 y Fk(\002)t Fg(R)p Fp(.)35 b(Th)n(us)20 b(there)h(is)f(no)g (geometric)g(con)n(tact)f(structure)i(on)f(this)h(geometry)-7 b(.)1278 855 y(3.)41 b Fo(Geometric)32 b(Engel)f(str)n(uctures)555 1004 y Fp(No)n(w)37 b(w)n(e)g(come)f(to)h(the)h(main)f(part)g(of)g (this)g(article,)i(the)f(classi\014cation)e(of)h(geometric)456 1104 y(Engel)27 b(structures)h(up)h(to)f(equiv)-5 b(alence.)39 b(Let)29 b(us)f(\014rst)g(mak)n(e)g(some)g(general)f(remarks)g(ab)r (out)456 1204 y(the)h(isometry)e(group)h(of)g(a)g(maximal)g(geometric)g (Engel)g(structure)g(\()p Fm(X)r(;)14 b Fk(D)r Fm(;)g(G)p Fp(\).)555 1303 y(Ev)n(ery)26 b(isometry)g(preserving)f(an)i(Engel)g (structure)f Fk(D)k Fp(has)c(to)h(preserv)n(e)f(the)h(subbundles)456 1450 y(\(8\))855 b Fk(W)30 b(\032)23 b(D)i(\032)e(E)30 b Fp(=)23 b([)p Fk(D)r Fm(;)14 b Fk(D)r Fp(])24 b Fk(\032)e Fm(T)12 b(M)36 b(:)456 1597 y Fp(Hence)d(the)h(iden)n(tit)n(y)f(comp)r (onen)n(t)g(of)h(the)g(stabilizer)e(of)h(a)g(p)r(oin)n(t)h Fm(x)f Fk(2)g Fm(X)40 b Fp(in)34 b(the)f(group)g(of)456 1697 y(isometries)28 b Fm(H)37 b Fp(preserving)28 b(an)h(Engel)g (structure)h Fk(D)i Fp(acts)d(trivially)g(on)h Fm(T)2816 1709 y Ff(x)2857 1697 y Fm(X)7 b Fp(.)43 b(This)29 b(implies)456 1797 y(that)24 b Fm(H)31 b Fp(has)24 b(dimension)h(four.)35 b(Since)25 b Fm(X)30 b Fp(is)25 b(simply)f(connected,)h Fm(G)g Fp(acts)e(freely)h(on)h Fm(X)7 b Fp(.)35 b(Th)n(us)456 1896 y(w)n(e)27 b(can)g(\014nd)h(automorphisms)e(of)i(Th)n(urston)f (geometries)f(as)h(in)h(Example)e(2.)555 1996 y(The)g(other)e (connected)i(comp)r(onen)n(ts)e(of)i(the)f(stabilizer)g(of)g Fm(x)h Fp(act)f(b)n(y)g(isometries)g(on)g Fm(T)3328 2008 y Ff(x)3369 1996 y Fm(X)456 2096 y Fp(and)19 b(they)i(preserv)n(e)d (\(8\).)34 b(F)-7 b(or)19 b(ev)n(ery)g(lo)r(cal)g(framing)h Fm(X)r(;)14 b(Y)38 b Fp(of)20 b Fk(D)i Fp(w)n(e)e(obtain)f(the)i(lo)r (cal)e(framing)456 2195 y Fm(X)r(;)14 b(Y)5 b(;)14 b Fp([)p Fm(X)r(;)g(Y)j Fp(])27 b(of)g Fk(E)7 b Fp(.)37 b(Note)26 b(that)h(the)g(orien)n(tation)f(of)g Fk(E)31 b Fp(=)22 b([)p Fk(D)r Fm(;)14 b Fk(D)r Fp(])28 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Fm(;)14 b Fp(Nil)2112 2967 y Fn(3)2163 3004 y Fg(nR)p Fp(\))83 b(\()2447 2982 y Fi(e)2435 3004 y Fp(Sl)q(\(2\))18 b Fk(\002)g Fg(R)p Fm(;)2821 2982 y Fi(e)2809 3004 y Fp(Sl\(2\))h Fk(\002)f Fg(R)p Fp(\))722 3108 y(\(Sol)865 3071 y Fn(4)902 3108 y Fp(\()p Fm(m;)c(n)p Fp(\))p Fm(;)g Fp(Sol)1274 3071 y Fn(4)1311 3108 y Fp(\()p Fm(m;)g(n)p Fp(\)\))84 b(\(Sol)1794 3071 y Fn(4)1794 3129 y(0)1831 3108 y Fm(;)14 b Fp(Sol)1979 3071 y Fn(4)2016 3108 y Fp(\()p Fm(\025)p Fp(\)\))243 b(\(Sol)2546 3071 y Fn(4)2546 3129 y(1)2583 3108 y Fm(;)14 b Fp(Sol)2731 3071 y Fn(4)2731 3129 y(1)2768 3108 y Fp(\))722 3213 y(\(Nil)863 3176 y Fn(4)900 3213 y Fm(;)g Fp(Nil)1045 3176 y Fn(4)1082 3213 y Fp(\))456 3356 y Fl(No)41 b(sub)l(ge)l(ometry)f(of)i(the)f(fol)t (lowing)i(Thurston)e(ge)l(ometries)g(admits)h(a)f(ge)l(ometric)h(Engel) 456 3456 y(structur)l(e:)857 3581 y Fm(S)913 3551 y Fn(4)1227 3581 y Fg(R)1287 3551 y Fn(4)1602 3581 y Fg(H)1667 3551 y Fn(4)2354 3581 y Fg(CP)2465 3551 y Fn(2)2743 3581 y Fg(HC)2868 3551 y Fn(2)857 3681 y Fm(S)913 3651 y Fn(2)968 3681 y Fk(\002)18 b Fm(S)1107 3651 y Fn(2)1227 3681 y Fm(S)1283 3651 y Fn(2)1339 3681 y Fk(\002)g Fg(R)1482 3651 y Fn(2)1602 3681 y Fm(S)1658 3651 y Fn(2)1713 3681 y Fk(\002)g Fg(H)1861 3651 y Fn(2)2354 3681 y Fg(H)2419 3651 y Fn(2)2475 3681 y Fk(\002)g Fg(H)2623 3651 y Fn(2)2743 3681 y Fg(H)2808 3651 y Fn(2)2863 3681 y Fk(\002)g Fg(R)3006 3651 y Fn(2)857 3785 y Fm(F)922 3755 y Fn(4)1227 3785 y Fg(H)1292 3755 y Fn(3)1348 3785 y Fk(\002)g Fg(R)111 b Fp(\(Nil)1743 3748 y Fn(3)1794 3785 y Fk(\002)p Fg(R)p Fm(;)14 b Fp(Nil)2064 3748 y Fn(3)2115 3785 y Fk(\002)p Fg(R)p Fp(\))555 3938 y(F)-7 b(or)32 b(the)i(pro)r(of)e(w)n(e)g (consider)g(the)h(individual)g(cases)f(where)g(w)n(e)h(also)f(discuss)g (the)h(non{)456 4038 y(maximal)25 b(Th)n(urston)f(geometries)g(lik)n(e) h(for)g(example)g(\(Sol)2303 4001 y Fn(4)2303 4058 y(0)2341 4038 y Fm(;)14 b Fp(Sol)2488 4001 y Fn(4)2525 4038 y Fp(\()p Fm(\025)p Fp(\)\).)38 b(W)-7 b(e)26 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b(NON{INTEGRABLE)i(PLANE)f(FIELDS)f(ON)h(THURSTON)h(GEOMETRIES)119 b(11)456 450 y Fp(3.2.)40 b Ft(Prolongation)35 b(of)g(geometric)f(con)m (tact)j(structures.)k Fp(Three)30 b(Th)n(urston)g(geome-)456 550 y(tries)j(of)i(dimension)f(4)g(can)f(b)r(e)i(treated)f(using)g(a)g (construction)f(similar)g(to)h(prolongation,)456 649 y(cf.)19 b(Example)f(6.)34 b(Recall)18 b(that)i(the)f(3{dimensional)e (Th)n(urston)i(geometries)e Fm(X)29 b Fp(=)23 b Fm(S)3072 619 y Fn(3)3109 649 y Fm(;)14 b Fp(Nil)3254 612 y Fn(3)3311 649 y Fp(and)467 734 y Fi(e)456 756 y Fp(Sl\(2\))31 b(admit)g (geometric)f(con)n(tact)h(structures)f(\()p Fm(X)r(;)14 b Fk(C)5 b Fm(;)14 b(H)7 b Fp(\))31 b(suc)n(h)g(that)g Fm(H)38 b Fp(is)31 b(4{dimensional,)456 855 y(connected)21 b(and)h(the)g(stabilizer)f(of)h(p)r(oin)n(ts)g(in)g Fm(X)28 b Fp(acts)21 b(b)n(y)h(rotations)e(on)i(the)g(con)n(tact)f(plane)h(at) 456 955 y(that)k(p)r(oin)n(t.)37 b(The)27 b(action)f(of)g Fm(H)34 b Fp(is)26 b(free)h(and)f(transitiv)n(e)g(on)g(the)h(manifold)f Fm(S)2921 925 y Fn(1)2958 955 y Fk(C)32 b Fp(of)26 b(unit)h(v)n(ec-)456 1055 y(tors)i(of)g(the)h(con)n(tact)f(structure.)43 b(The)30 b(action)f(is)h(isometric)f(for)g(the)h(induced)h(Riemannian)456 1154 y(metric)c(on)g Fm(S)885 1124 y Fn(1)922 1154 y Fk(C)5 b Fp(.)555 1254 y(Lifting)32 b(all)g(data)f(to)h(the)h(univ)n (ersal)d(co)n(v)n(erings)f(w)n(e)j(obtain)g(an)f(isometric)g(group)g (action)456 1366 y(of)580 1345 y Fi(e)561 1366 y Fm(H)45 b Fp(on)812 1341 y Fi(g)801 1366 y Fm(S)857 1342 y Fn(1)894 1366 y Fk(C)e Fp(whic)n(h)c(is)f(free)g(and)h(transitiv)n(e.)68 b(In)39 b(particular,)h(the)e(preimage)g(in)3283 1345 y Fi(e)3263 1366 y Fm(H)46 b Fp(of)456 1472 y(a)36 b(co)r(compact)g (lattice)g(\000)i Fk(\032)g Fm(H)43 b Fp(is)37 b(a)f(co)r(compact)g (lattice)2381 1451 y Fi(e)2378 1472 y Fp(\000)h(in)2592 1451 y Fi(e)2573 1472 y Fm(H)7 b Fp(.)64 b(Th)n(us)36 b(\()p Fm(S)3043 1441 y Fn(1)3080 1472 y Fk(C)5 b Fm(;)3185 1451 y Fi(e)3166 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Fh(sl)p Fp(\(2)p Fm(;)14 b Fg(R)p Fp(\))22 b Fk(\010)g Fg(R)33 b Fp(and)456 3993 y(this)27 b(con)n(tradicts)g (that)h Fk(D)1291 3963 y Fj(0)1337 3993 y Fk(\033)23 b(W)1514 3963 y Fj(0)1565 3993 y Fp(is)k(an)h(Engel)e(structure.)555 4100 y(The)32 b(c)n(haracteristic)e(foliation)i(of)g(eac)n(h)f (geometric)g(Engel)g(structure)h Fk(D)2911 4069 y Fj(0)2967 4100 y Fp(on)3098 4078 y Fi(e)3086 4100 y Fp(Sl\(2)p Fm(;)14 b Fg(R)p Fp(\))22 b Fk(\002)456 4199 y Fg(R)38 b Fp(is)h(tangen)n(t)f(to)g(the)i(\014rst)e(factor.)69 b(This)39 b(implies)g(that)g Fk(E)2450 4169 y Fj(0)2515 4199 y Fp(=)i([)p Fk(D)2710 4169 y Fj(0)2733 4199 y Fm(;)14 b Fk(D)r Fp(])39 b(induces)g(a)f(left{)456 4299 y(in)n(v)-5 b(arian)n(t)28 b(foliation)h(on)g(the)g(\014rst)g(factor.)41 b(F)-7 b(rom)29 b(the)h(description)f(of)g(the)h(coadjoin)n(t)e(orbits) 456 4405 y(in)21 b Fh(sl)606 4368 y Fj(\003)644 4405 y Fp(\(2)p Fm(;)14 b Fg(R)p Fp(\))21 b(in)h(Section)f(2.3)f(one)h(can)g (read)f(of)h(that)h(all)f(left{in)n(v)-5 b(arian)n(t)20 b(foliations)g(on)3184 4383 y Fi(e)3172 4405 y Fp(Sl\(2)p Fm(;)14 b Fg(R)p Fp(\))456 4505 y(are)26 b(conjugate)h(to)h(eac)n(h)e (other.)555 4611 y(Hence)32 b(w)n(e)f(ma)n(y)g(assume)g(that)h Fk(E)1642 4581 y Fj(0)1687 4611 y Fk(\\)1775 4589 y Fi(e)1763 4611 y Fp(Sl)q(\(2)p Fm(;)14 b Fg(R)p Fp(\))31 b(is)h(spanned)f(b)n(y)g Fm(A;)14 b(C)39 b Fp(and)31 b(that)h Fm(@)3191 4623 y Ff(t)3242 4611 y Fp(+)20 b Fm(\026B)456 4711 y Fp(extends)29 b(this)g(to)g(a)f(framing)g(of)h Fk(E)1551 4681 y Fj(0)1604 4711 y Fp(with)g Fm(\026)d Fk(2)f Fg(R)p Fp(.)41 b(If)30 b Fm(\026)25 b Fp(=)g(0,)k(then)g Fk(E)2659 4681 y Fj(0)2712 4711 y Fp(is)g(a)f(foliation)h(and)g(no)456 4810 y(ev)n(en)23 b(con)n(tact)g(structure.)35 b(After)24 b(applying)f(an)g(automorphism) f(of)i(the)g(Th)n(urston)f(geometry)456 4917 y(\()500 4895 y Fi(e)488 4917 y Fp(Sl\(2)p Fm(;)14 b Fg(R)p Fp(\))19 b Fk(\002)f Fg(R)p Fm(;)971 4895 y Fi(e)959 4917 y Fp(Sl\(2)p Fm(;)c Fg(R)p Fp(\))19 b Fk(\002)g Fg(R)p Fp(\))28 b(similar)g(to)g (\(10\))g(w)n(e)f(ma)n(y)h(assume)f Fm(\026)e Fp(=)e(1.)39 b(In)28 b(this)h(case,)e(the)456 5016 y(c)n(haracteristic)e(foliation)i (of)h Fk(E)1437 4986 y Fj(0)1488 5016 y Fp(is)g(spanned)f(b)n(y)g Fm(A)p Fp(.)555 5116 y(So)j(there)g(is)g Fm(\027)i Fk(2)27 b Fg(R)j Fp(suc)n(h)g(that)g(the)h(Engel)e(structure)g(is)h(spanned)g (b)n(y)g Fm(A)g Fp(and)g Fm(@)3159 5128 y Ff(t)3208 5116 y Fp(+)20 b Fm(B)k Fp(+)456 5216 y Fm(\027)5 b(C)h Fp(.)37 b(Finally)-7 b(,)26 b(conjugation)g(with)h(elemen)n(ts)f(of)h(exp\()p Fg(R)p Fm(A)p Fp(\))g(sho)n(ws)e(that)i(all)f(geometric)f(Engel)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 456 251 a Fn(14)1266 b(T.)23 b(V)n(OGEL)456 450 y Fp(structures)h(on)g(\()998 428 y Fi(e)986 450 y Fp(Sl)q(\(2)p Fm(;)14 b Fg(R)p Fp(\))f Fk(\002)g Fg(R)p Fm(;)1458 428 y Fi(e)1447 450 y Fp(Sl)o(\(2)p Fm(;)h Fg(R)p Fp(\))f Fk(\002)g Fg(R)p Fp(\))25 b(are)f(equiv)-5 b(alen)n(t)25 b(to)g Fk(D)2617 462 y Fn(0)2679 450 y Fp(with)g Fm(\027)k Fp(=)22 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b(the)h(maximal)f(group)f(of)i(Engel)f(preserving) e(isome-)456 1229 y(tries)j(has)g(four)g(comp)r(onen)n(ts,)g(w)n(e)h (ha)n(v)n(e)e(seen)h(b)r(efore)g(that)h(there)g(cannot)f(b)r(e)h(more)e (of)i(them.)456 1379 y(3.3.)40 b Ft(Solv)-5 b(able)32 b(Geometries.)39 b Fp(In)29 b(this)g(section)f(w)n(e)g(sho)n(w)f(that)i (eac)n(h)f(of)g(the)h(solv)-5 b(able)27 b(ge-)456 1479 y(ometries)36 b(Sol)909 1441 y Fn(4)946 1479 y Fp(\()p Fm(m;)14 b(n)p Fp(\))p Fm(;)g Fp(Sol)1318 1441 y Fn(4)1318 1499 y(1)1392 1479 y Fp(and)37 b(all)g(subgeometries)f(of)i(Sol)2449 1441 y Fn(4)2449 1499 y(0)2524 1479 y Fp(admit)f(a)g(geometric)f(Engel) 456 1578 y(structure)26 b(whic)n(h)h(is)g(unique)g(up)g(to)g(equiv)-5 b(alence.)37 b(Then)27 b(w)n(e)f(compare)g(theses)h(Engel)f(struc-)456 1678 y(tures)h(with)h(the)g(construction)f(of)g(Geiges)g(describ)r(ed)h (in)f(Example)g(7.)456 1828 y(3.3.1.)40 b Fm(X)29 b Fp(=)23 b(Sol)988 1791 y Fn(4)1025 1828 y Fp(\()p Fm(m;)14 b(n)p Fp(\))p Fl(.)42 b Fp(Let)28 b Fm(m;)14 b(n)27 b Fp(b)r(e)h(p)r(ositiv)n (e)f(in)n(tegers)g(suc)n(h)g(that)h(the)g(zero)r(es)e(of)456 1964 y(\(12\))770 b Fm(P)12 b Fp(\()p Fm(m;)i(n)p Fp(\))23 b(=)f Fk(\000)p Fm(\025)1886 1929 y Fn(3)1942 1964 y Fp(+)c Fm(m\025)2146 1929 y Fn(2)2202 1964 y Fk(\000)g Fm(n\025)h Fp(+)f(1)456 2099 y(are)29 b(pairwise)h(di\013eren)n(t)h (real)f(n)n(um)n(b)r(ers)g Fm(e)1797 2068 y Ff(\013)1844 2099 y Fm(;)14 b(e)1920 2068 y Ff(\014)1964 2099 y Fm(;)g(e)2040 2068 y Ff(\015)2113 2099 y Fp(with)31 b Fm(\013)21 b Fp(+)f Fm(\014)25 b Fp(+)20 b Fm(\015)33 b Fp(=)28 b(0)i(and)h Fm(\013)d(>)g(\014)33 b(>)28 b(\015)5 b Fp(.)456 2198 y(Other)23 b(p)r(ossible)h(con\014gurations)f(of)h(the)h(zero)r(es)e (of)h Fm(P)12 b Fp(\()p Fm(m;)i(n)p Fp(\))25 b(will)f(b)r(e)h (discussed)f(b)r(elo)n(w.)35 b(The)456 2302 y(Lie)27 b(group)g(Sol)941 2265 y Fn(4)979 2302 y Fp(\()p Fm(m;)14 b(n)p Fp(\))27 b(is)h(the)g(semidirect)f(pro)r(duct)h Fg(R)2226 2272 y Fn(3)2281 2302 y Fg(n)18 b(R)28 b Fp(with)g Fg(R)g 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y(to)c(the)i(\014b)r(ers)e(of)h(the)g(susp)r(ension)g(describ)r (ed)f(in)h(\(29\).)34 b(Th)n(us)20 b(the)g(geometric)f(Engel)g (structure)456 2571 y(on)27 b(\000)p Fk(n)14 b Fp(Nil)787 2534 y Fn(4)851 2571 y Fp(a)28 b(mo)r(di\014cation)f(of)h(the)g (construction)e(of)i(Geiges)f(with)h Fm(k)e Fp(=)c(0.)456 2798 y(3.5.)40 b Fm(X)30 b Fp(=)24 b Fg(R)874 2768 y Fn(4)911 2798 y Ft(.)41 b Fp(All)28 b(subgroups)f(of)h(the)g(isometry)f (group)g(of)h Fg(R)2478 2768 y Fn(4)2543 2798 y Fp(whic)n(h)g(act)g (transitiv)n(ely)f(on)456 2898 y Fg(R)516 2868 y Fn(4)580 2898 y Fp(m)n(ust)f(con)n(tain)g(the)h(translations)f(of)g Fg(R)1824 2868 y Fn(4)1861 2898 y Fp(.)37 b(The)27 b(only)f (translation)f(in)n(v)-5 b(arian)n(t)26 b(plane)g(\014elds)456 2997 y(on)h Fg(R)631 2967 y Fn(4)696 2997 y Fp(are)f(foliations.)37 b(Th)n(us)27 b(there)g(is)h(no)f(geometric)g(Engel)f(structure)h(on)h Fg(R)2988 2967 y Fn(4)3025 2997 y Fp(.)456 3224 y(3.6.)40 b Fm(X)46 b Fp(=)39 b Fg(H)910 3194 y Fn(3)971 3224 y Fk(\002)25 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b(of)g(integral)f(cur) -7 b(ves)555 600 y Fp(In)31 b(this)g(\014nal)f(section)g(w)n(e)g (consider)f(in)n(tegral)h(curv)n(es)f(of)h(an)g(Engel)g(structure)g Fk(D)j Fp(on)d Fm(M)9 b Fp(,)456 699 y(i.e.)47 b(w)n(e)30 b(consider)g(curv)n(es)g Fm(\015)j Fp(:)c([)p Fm(a;)14 b(b)p Fp(])28 b Fk(\000)-14 b(!)29 b Fm(M)39 b Fp(suc)n(h)31 b(that)44 b(_)-36 b Fm(\015)5 b Fp(\()p Fm(t)p Fp(\))29 b Fk(2)g(D)r Fp(.)48 b(W)-7 b(e)31 b(denote)g(the)g(space)456 799 y(of)25 b(all)h(in)n(tegral)f(curv)n(es)f(of)i Fk(D)i Fp(starting)d(at)h Fm(p)g Fp(and)f(ending)h(at)g Fm(q)j Fp(b)n(y)c(\012)2640 811 y Fj(D)2698 799 y Fp(\()p Fm(p;)14 b(q)s Fp(\))26 b(and)g(w)n(e)f(use)h(the)456 898 y Fm(C)521 868 y Fn(1)558 898 y Fp({top)r(ology)g(on)h(\012)1116 910 y Fj(D)1174 898 y Fp(\()p Fm(p;)14 b(q)s Fp(\).)456 1057 y Ft(De\014nition)46 b(13.)j Fp(An)41 b(in)n(tegral)f(curv)n(e)g Fm(\015)51 b Fp(:)45 b([)p Fm(a;)14 b(b)p Fp(])46 b Fk(\000)-15 b(!)46 b Fm(M)k Fp(of)41 b Fk(D)i Fp(is)e Fl(rigid)i 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Fp(\))30 b(=)f Fk(C)c(\010)c(W)7 b Fp(\()p Fm(p)p Fp(\).)48 b(F)-7 b(or)31 b Fm(q)j Fp(in)e Fk(W)7 b Fp(\()p Fm(p)p Fp(\))31 b(let)h Fm(')3291 1725 y Ff(q)3359 1713 y Fp(b)r(e)456 1813 y(germ)27 b(of)h(the)h(holonom)n (y)e(of)h Fk(W)35 b Fp(mapping)28 b(a)g(neigh)n(b)r(ourho)r(o)r(d)f(of) h Fm(q)j Fp(to)e(a)e(neigh)n(b)r(ourho)r(o)r(d)h(of)456 1913 y Fm(p)p Fp(.)35 b(Recall)22 b(that)i(the)f(holonom)n(y)e(of)i Fk(W)30 b Fp(preserv)n(es)21 b(b)r(oth)i Fk(W)30 b Fp(and)23 b Fk(E)7 b Fp(.)36 b(Therefore)21 b(the)j(follo)n(wing)456 2012 y(de\014nition)k(of)f(the)h Fl(development)j(map)e Fm(\016)1753 2024 y Ff(p)1819 2012 y Fp(mak)n(es)d(sense)1414 2167 y Fm(\016)1451 2179 y Ff(p)1513 2167 y Fp(:)d Fk(W)7 b Fp(\()p Fm(p)p Fp(\))23 b Fk(\000)-14 b(!)23 b Fg(P)p Fk(C)k(')c Fg(RP)2255 2132 y Fn(1)1714 2296 y Fm(q)j Fk(7\000)-14 b(!)23 b Fp(\()p Fm(')2020 2308 y Ff(q)r Fj(\003)2091 2296 y Fk(D)r Fp(\()p Fm(q)s Fp(\)\))d Fk(\\)e(C)33 b Fm(:)456 2451 y Fp(The)24 b(condition)f(that)h([)p Fk(D)r 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