- Commit
- c93e7373196dc96e6b9d65275f85ce3b4dffcc59
- Parent
- 2dc7ea2f5adfbd9db880ef1b9864a09597635d7d
- Author
- Pablo <pablo-pie@riseup.net>
- Date
Changed the notation for the number of boundary components
b → p
memoire-m2
My M2 Memoire on mapping class groups & their representations
Diffstat
7 files changed, 182 insertions, 177 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Modified | images/lickorish-gens-gen-2.svg | 35 | 18 | 17 |
| Modified | images/lickorish-gens-korkmaz-proof.svg | 33 | 17 | 16 |
| Modified | images/lickorish-gens.svg | 35 | 19 | 16 |
| Modified | sections/introduction.tex | 14 | 7 | 7 |
| Modified | sections/presentation.tex | 92 | 46 | 46 |
| Modified | sections/representations.tex | 102 | 51 | 51 |
| Modified | sections/twists.tex | 48 | 24 | 24 |
diff --git a/images/lickorish-gens-gen-2.svg b/images/lickorish-gens-gen-2.svg @@ -24,13 +24,13 @@ inkscape:deskcolor="#d1d1d1" inkscape:document-units="mm" showguides="false" - inkscape:zoom="1.4142136" - inkscape:cx="105.71246" - inkscape:cy="18.384776" - inkscape:window-width="1358" - inkscape:window-height="728" - inkscape:window-x="4" - inkscape:window-y="36" + inkscape:zoom="2.0000001" + inkscape:cx="184.75" + inkscape:cy="111" + inkscape:window-width="1366" + inkscape:window-height="768" + inkscape:window-x="0" + inkscape:window-y="0" inkscape:window-maximized="1" inkscape:current-layer="layer1"> <sodipodi:guide @@ -143,20 +143,12 @@ d="m 145.99197,170.62155 c -0.0154,0.46245 -0.025,0.92504 -0.0324,1.38768 -0.004,0.25842 -0.007,0.51685 -0.009,0.77529 -0.002,0.31367 -0.0133,0.17892 0.7942,0.0845 0.0391,-0.005 0.0133,-0.0776 0.0199,-0.11639 0.0631,-0.35486 0.13788,-0.7139 0.32404,-1.0266 0.0169,-0.018 0.0317,-0.0383 0.0507,-0.054 0.0101,-0.008 0.0222,-0.0162 0.0352,-0.017 0.005,-2.7e-4 0.009,0.0116 0.005,0.0145 -0.0689,0.0483 -0.14623,0.0163 -0.1624,0.0741 0.14757,0.55425 0.0937,1.14167 0.0695,1.70835 -0.0262,0.71116 -0.10631,1.4495 0.0742,2.14932 0.033,0.12788 0.0864,0.2496 0.12966,0.3744 0.20445,0.44055 0.54085,0.86479 1.09519,0.62575 0.22675,-0.0978 0.34934,-0.30753 0.48604,-0.4925 0.10808,-0.1869 0.23793,-0.37539 0.28902,-0.58933 0.0143,-0.0598 0.0157,-0.12198 0.0236,-0.18298 0.0244,-0.56073 -0.76864,-0.59519 -0.79301,-0.0345 v 0 c 0.006,-0.0119 0.019,-0.0487 0.0171,-0.0356 -0.0204,0.14066 -0.12444,0.26076 -0.18441,0.38359 -0.0325,0.0497 -0.12802,0.16383 -0.0995,0.20174 0.002,0.002 0.22865,0.11101 0.10029,0.10261 -0.0869,-0.006 -0.18237,-0.23923 -0.18823,-0.2497 -0.0314,-0.0807 -0.0716,-0.15853 -0.0942,-0.24214 -0.17368,-0.64186 -0.0812,-1.32446 -0.0627,-1.97701 0.0246,-0.58436 0.0978,-1.3914 -0.10353,-1.94666 -0.0355,-0.098 -0.0885,-0.18868 -0.13272,-0.28302 -0.1684,-0.18241 -0.21616,-0.29268 -0.48321,-0.33979 -0.33351,-0.0588 -0.58676,0.17062 -0.76157,0.41749 -0.22116,0.39947 -0.34649,0.83482 -0.41654,1.28553 -0.007,0.0435 -0.0647,0.11844 -0.0222,0.13064 0.80765,0.23211 0.7827,0.33742 0.78427,0.0402 0.002,-0.25599 0.005,-0.51198 0.009,-0.76796 0.007,-0.46164 0.0171,-0.92323 0.0324,-1.38469 0.0111,-0.56116 -0.78245,-0.57691 -0.79359,-0.0157 z" /> <path style="fill:#f0c03b;fill-opacity:1;stroke-width:10.5833;stroke-linejoin:bevel" - id="path89" - d="m 149.77933,175.53002 c -0.0142,0.47726 -0.0234,0.95461 -0.0306,1.43202 -0.01,0.34661 0.005,0.69297 0.0126,1.03937 -4e-5,-0.0123 -8e-5,-0.0247 -1.2e-4,-0.037 -0.0797,0.55557 0.70595,0.66834 0.7857,0.11277 v 0 c 0.003,-0.0311 0.005,-0.0622 0.008,-0.0933 -0.008,-0.33654 -0.0215,-0.67303 -0.0125,-1.00978 0.007,-0.47626 0.0164,-0.95247 0.0306,-1.42857 0.011,-0.56116 -0.78266,-0.57664 -0.7936,-0.0155 z" /> - <path - style="fill:#f0c03b;fill-opacity:1;stroke-width:10.5833;stroke-linejoin:bevel" - id="path90" - d="m 150.63956,177.44402 c -0.0149,-0.0561 0.19266,-0.19947 0.12272,-0.10761 -0.0117,0.01 -0.0376,0.0142 -0.035,0.0291 0.004,0.0235 0.0335,0.0341 0.049,0.0522 0.021,0.0247 0.044,0.0483 0.0604,0.0764 0.035,0.0599 0.0298,0.12423 0.0213,0.18912 -0.0179,0.0778 -0.0408,0.17388 -0.12443,0.20273 -0.0304,10e-4 -0.06,0.005 -0.0905,0.002 -0.0195,-0.002 -0.0823,-0.0119 0.007,0.002 -0.5387,-0.15754 -0.76148,0.60431 -0.22278,0.76184 v 0 c 0.0687,0.0156 0.13493,0.0305 0.20579,0.0299 0.19738,-2.6e-4 0.3872,-0.0392 0.55659,-0.14641 0.29001,-0.21872 0.44666,-0.51482 0.46171,-0.882 -0.006,-0.0776 -0.003,-0.1564 -0.0185,-0.23273 -0.0641,-0.32323 -0.29454,-0.6535 -0.60325,-0.7895 -0.10053,-0.0443 -0.21185,-0.0582 -0.31778,-0.0873 -0.0897,0.0153 -0.1825,0.0182 -0.26921,0.046 -0.24225,0.0775 -0.41094,0.30086 -0.52138,0.51647 -0.23849,0.50808 0.48004,0.84535 0.71853,0.33727 z" /> - <path - style="fill:#f0c03b;fill-opacity:1;stroke-width:10.5833;stroke-linejoin:bevel" id="path91" - d="m 152.2881,177.69871 c 0.0368,5.3e-4 0.0735,7.9e-4 0.11028,0.001 0.11883,10e-4 0.23766,0.002 0.35649,0.003 0.0938,7.9e-4 0.11844,-0.008 0.078,0.004 0.5527,0.0977 0.69084,-0.68396 0.13814,-0.78164 v 0 c -0.22266,-0.029 -0.44772,-0.0138 -0.67189,-0.0209 -0.56121,-0.008 -0.5722,0.7859 -0.011,0.79367 z" /> + d="m 152.55418,175.77149 c 0.0368,5.3e-4 0.0735,7.9e-4 0.11028,0.001 0.11883,0.001 0.23766,0.002 0.35649,0.003 0.0938,7.9e-4 0.11844,-0.008 0.078,0.004 0.5527,0.0977 0.69084,-0.68396 0.13814,-0.78164 v 0 c -0.22266,-0.029 -0.44772,-0.0138 -0.67189,-0.0209 -0.56121,-0.008 -0.5722,0.7859 -0.011,0.79367 z" /> <path style="fill:#f0c03b;fill-opacity:1;stroke-width:10.5833;stroke-linejoin:bevel" id="path92" - d="m 154.36864,177.1562 c 0.0722,-0.21667 0.16845,-0.4247 0.26973,-0.6291 0.076,-0.15333 0.0392,-0.10883 0.10443,-0.1776 -0.19793,0.0192 -0.39586,0.0383 -0.59378,0.0575 -0.022,-0.0164 -0.0563,-0.075 -0.0659,-0.0493 -0.0252,0.0677 0.003,0.1445 0.004,0.21676 0.005,0.32675 0.017,0.65166 0.0358,0.97793 0.0266,0.30078 0.0331,0.60691 0.0951,0.90357 0.12576,0.547 0.89933,0.36916 0.77358,-0.17784 v 0 c 0.002,0.0117 0.0132,0.0958 -0.007,-0.0423 -0.005,-0.0343 -0.0102,-0.0686 -0.0144,-0.10299 -0.0254,-0.20801 -0.0418,-0.41709 -0.0546,-0.62621 -0.0145,-0.2499 -0.0243,-0.38597 -0.0309,-0.63841 -8e-4,-0.0302 -0.005,-0.55414 -0.012,-0.59842 -0.0205,-0.12109 0.009,-0.29885 -0.1,-0.35459 -0.19874,-0.10125 -0.44542,-0.0245 -0.66813,-0.0367 -0.22182,0.32097 -0.36781,0.69092 -0.49774,1.05711 -0.156,0.53915 0.60648,0.75977 0.76248,0.22061 z" /> + d="m 154.63472,175.22898 c 0.0722,-0.21667 0.16845,-0.4247 0.26973,-0.6291 0.076,-0.15333 0.0392,-0.10883 0.10443,-0.1776 -0.19793,0.0192 -0.39586,0.0383 -0.59378,0.0575 -0.022,-0.0164 -0.0563,-0.075 -0.0659,-0.0493 -0.0252,0.0677 0.003,0.1445 0.004,0.21676 0.005,0.32675 0.017,0.65166 0.0358,0.97793 0.0266,0.30078 0.0331,0.60691 0.0951,0.90357 0.12576,0.547 0.89933,0.36916 0.77358,-0.17784 v 0 c 0.002,0.0117 0.0132,0.0958 -0.007,-0.0423 -0.005,-0.0343 -0.0102,-0.0686 -0.0144,-0.10299 -0.0254,-0.20801 -0.0418,-0.41709 -0.0546,-0.62621 -0.0145,-0.2499 -0.0243,-0.38597 -0.0309,-0.63841 -8e-4,-0.0302 -0.005,-0.55414 -0.012,-0.59842 -0.0205,-0.12109 0.009,-0.29885 -0.1,-0.35459 -0.19874,-0.10125 -0.44542,-0.0245 -0.66813,-0.0367 -0.22182,0.32097 -0.36781,0.69092 -0.49774,1.05711 -0.156,0.53915 0.60648,0.75977 0.76248,0.22061 z" /> <path id="path45" style="fill:#568bb5;fill-opacity:1;stroke-width:10.5833;stroke-linejoin:bevel" @@ -181,5 +173,14 @@ id="path7" d="m 135.03923,169.81766 c 0.42606,-0.0419 0.038,-0.0673 0.0572,-0.1013 -0.0103,-0.004 -0.0223,0.005 -0.0303,0.0122 -0.0501,0.0461 -0.0874,0.10465 -0.13454,0.15376 -0.0797,0.083 -0.10211,0.10111 -0.18442,0.17629 -0.20939,0.19865 -0.30213,0.39003 -0.10721,0.70777 0.12893,0.21017 0.52385,0.18632 0.72777,0.19301 0.29695,0.003 0.59391,-0.003 0.89069,-0.0131 0.0176,-5.3e-4 0.0352,-0.001 0.0528,-0.002 -0.001,2.6e-4 -0.003,7.9e-4 -0.004,0.001 0.56064,0.0264 0.59802,-0.76644 0.0374,-0.79287 v 0 c -0.0784,-0.002 -0.0411,-0.002 -0.11194,2.6e-4 -0.28047,0.009 -0.56109,0.0153 -0.84173,0.013 -0.0536,-0.001 -0.10725,-0.003 -0.16087,-0.004 -0.0271,-5.3e-4 -0.0884,-0.0279 -0.0814,-0.002 0.0132,0.0496 0.0968,0.0583 0.10971,0.10801 0.0318,0.12285 0.0271,0.25386 0.0167,0.38033 -0.002,0.0256 -0.0746,0.0696 -0.0528,0.056 0.0249,-0.0155 0.0417,-0.0414 0.0626,-0.0621 0.26616,-0.24626 0.58865,-0.54931 0.6055,-0.94059 0.005,-0.10743 -0.0323,-0.21263 -0.0485,-0.31894 -0.0635,-0.0859 -0.10908,-0.18843 -0.19047,-0.25754 -0.29794,-0.25304 -0.7102,-0.15536 -1.01722,0.01 -0.48254,0.28665 -0.0772,0.96908 0.40538,0.68243 z" sodipodi:nodetypes="cccccsccccsscccccccccccccc" /> + <path + style="fill:#f0c03b;fill-opacity:1;stroke-width:10.5833;stroke-linejoin:bevel" + id="path9" + d="m 150.67463,174.88225 c -0.006,-0.0655 -0.002,-0.13456 -0.0205,-0.19832 -0.0115,-0.0387 -0.0269,-0.0762 -0.0404,-0.11431 -0.24499,0.0198 -0.35562,0.17065 -0.59368,0.23195 -0.0177,0.005 -0.1903,0.10898 -0.1906,0.11177 -0.0334,0.30784 -0.0396,0.61576 -0.0469,0.92509 -0.008,0.70937 0.0286,1.41932 0.003,2.12849 -0.004,0.032 -0.007,0.0641 -0.0109,0.0961 -0.0581,0.55825 0.73137,0.64044 0.78948,0.0822 v 0 c 0.005,-0.05 0.01,-0.1 0.0147,-0.14998 0.0248,-0.71255 -0.008,-1.42568 -0.002,-2.13843 0.006,-0.27878 0.005,-0.38819 0.0234,-0.65433 0.004,-0.0634 0.011,-0.1267 0.0176,-0.18995 0.004,-0.0373 0.0114,-0.0742 0.0136,-0.11158 1.5e-4,-0.003 -0.005,0.002 -0.007,0.003 -0.24006,0.006 -0.48013,0.0129 -0.72019,0.0193 -0.01,-0.0303 -0.0192,-0.0606 -0.0287,-0.0909 0.002,0.004 0.003,0.005 0.003,0.0109 0.003,0.0321 0.003,0.0645 0.005,0.0967 0.0408,0.55978 0.83245,0.50207 0.79165,-0.0577 z" + sodipodi:nodetypes="ccccccccsscccccccccccc" /> + <path + style="fill:#f0c03b;fill-opacity:1;stroke-width:10.5833;stroke-linejoin:bevel" + id="path10" + d="m 150.48816,175.18442 c -0.007,0.0239 -0.0279,0.0394 -0.0387,0.0608 -0.002,0.003 0.006,-0.002 0.009,-0.004 0.01,-0.006 0.0191,-0.0123 0.029,-0.018 0.0412,-0.0239 0.0365,-0.0204 0.0797,-0.0396 0.0663,-0.0174 0.14088,-0.0385 0.20965,-0.0202 0.0132,0.004 0.0453,0.0297 0.0546,0.0368 0.0258,0.0265 0.0743,0.0765 0.0354,0.11156 -0.002,0.002 -0.004,0.005 -0.007,0.004 -0.002,-0.002 -0.007,-0.008 -0.0108,-0.008 -0.0156,-8e-4 -0.0559,0.0344 -0.063,0.0393 -0.0915,0.058 -0.1922,0.0979 -0.29654,0.12541 -0.0215,0.005 -0.0264,0.006 -0.0458,0.01 -0.003,5.3e-4 -0.0117,0.002 -0.009,10e-4 0.0315,-0.006 0.0243,-0.003 0.0731,-0.004 -0.56001,-0.0376 -0.61318,0.75437 -0.0532,0.79197 v 0 c 0.0839,-2.7e-4 0.16515,-0.01 0.24591,-0.0341 0.20001,-0.0564 0.39276,-0.13639 0.56342,-0.25694 0.0309,-0.0241 0.0631,-0.0464 0.0926,-0.0721 0.18323,-0.15981 0.2857,-0.35218 0.30334,-0.59598 -0.006,-0.13547 0.002,-0.18015 -0.0439,-0.31231 -0.0571,-0.16549 -0.16181,-0.31041 -0.29843,-0.41957 -0.0473,-0.0378 -0.10098,-0.0668 -0.15147,-0.10024 -0.0578,-0.0242 -0.11351,-0.0544 -0.1735,-0.0726 -0.25573,-0.0777 -0.52373,-0.0387 -0.76738,0.0588 -0.17157,0.0841 -0.34482,0.18147 -0.44272,0.35294 -0.2578,0.49856 0.44727,0.86314 0.70506,0.36458 z" /> </g> </svg>
diff --git a/images/lickorish-gens-korkmaz-proof.svg b/images/lickorish-gens-korkmaz-proof.svg @@ -25,12 +25,12 @@ inkscape:document-units="mm" showguides="false" inkscape:zoom="1.4142136" - inkscape:cx="261.27596" - inkscape:cy="147.07821" - inkscape:window-width="1358" - inkscape:window-height="728" - inkscape:window-x="4" - inkscape:window-y="36" + inkscape:cx="347.54297" + inkscape:cy="130.10764" + inkscape:window-width="1366" + inkscape:window-height="768" + inkscape:window-x="0" + inkscape:window-y="0" inkscape:window-maximized="1" inkscape:current-layer="layer1"> <sodipodi:guide @@ -256,20 +256,12 @@ d="m 145.99197,170.62155 c -0.0154,0.46245 -0.025,0.92504 -0.0324,1.38768 -0.004,0.25842 -0.007,0.51685 -0.009,0.77529 -0.002,0.31367 -0.0133,0.17892 0.7942,0.0845 0.0391,-0.005 0.0133,-0.0776 0.0199,-0.11639 0.0631,-0.35486 0.13788,-0.7139 0.32404,-1.0266 0.0169,-0.018 0.0317,-0.0383 0.0507,-0.054 0.0101,-0.008 0.0222,-0.0162 0.0352,-0.017 0.005,-2.7e-4 0.009,0.0116 0.005,0.0145 -0.0689,0.0483 -0.14623,0.0163 -0.1624,0.0741 0.14757,0.55425 0.0937,1.14167 0.0695,1.70835 -0.0262,0.71116 -0.10631,1.4495 0.0742,2.14932 0.033,0.12788 0.0864,0.2496 0.12966,0.3744 0.20445,0.44055 0.54085,0.86479 1.09519,0.62575 0.22675,-0.0978 0.34934,-0.30753 0.48604,-0.4925 0.10808,-0.1869 0.23793,-0.37539 0.28902,-0.58933 0.0143,-0.0598 0.0157,-0.12198 0.0236,-0.18298 0.0244,-0.56073 -0.76864,-0.59519 -0.79301,-0.0345 v 0 c 0.006,-0.0119 0.019,-0.0487 0.0171,-0.0356 -0.0204,0.14066 -0.12444,0.26076 -0.18441,0.38359 -0.0325,0.0497 -0.12802,0.16383 -0.0995,0.20174 0.002,0.002 0.22865,0.11101 0.10029,0.10261 -0.0869,-0.006 -0.18237,-0.23923 -0.18823,-0.2497 -0.0314,-0.0807 -0.0716,-0.15853 -0.0942,-0.24214 -0.17368,-0.64186 -0.0812,-1.32446 -0.0627,-1.97701 0.0246,-0.58436 0.0978,-1.3914 -0.10353,-1.94666 -0.0355,-0.098 -0.0885,-0.18868 -0.13272,-0.28302 -0.1684,-0.18241 -0.21616,-0.29268 -0.48321,-0.33979 -0.33351,-0.0588 -0.58676,0.17062 -0.76157,0.41749 -0.22116,0.39947 -0.34649,0.83482 -0.41654,1.28553 -0.007,0.0435 -0.0647,0.11844 -0.0222,0.13064 0.80765,0.23211 0.7827,0.33742 0.78427,0.0402 0.002,-0.25599 0.005,-0.51198 0.009,-0.76796 0.007,-0.46164 0.0171,-0.92323 0.0324,-1.38469 0.0111,-0.56116 -0.78245,-0.57691 -0.79359,-0.0157 z" /> <path style="fill:#f0c03b;fill-opacity:1;stroke-width:10.5833;stroke-linejoin:bevel" - id="path89" - d="m 149.77933,175.53002 c -0.0142,0.47726 -0.0234,0.95461 -0.0306,1.43202 -0.01,0.34661 0.005,0.69297 0.0126,1.03937 -4e-5,-0.0123 -8e-5,-0.0247 -1.2e-4,-0.037 -0.0797,0.55557 0.70595,0.66834 0.7857,0.11277 v 0 c 0.003,-0.0311 0.005,-0.0622 0.008,-0.0933 -0.008,-0.33654 -0.0215,-0.67303 -0.0125,-1.00978 0.007,-0.47626 0.0164,-0.95247 0.0306,-1.42857 0.011,-0.56116 -0.78266,-0.57664 -0.7936,-0.0155 z" /> - <path - style="fill:#f0c03b;fill-opacity:1;stroke-width:10.5833;stroke-linejoin:bevel" - id="path90" - d="m 150.63956,177.44402 c -0.0149,-0.0561 0.19266,-0.19947 0.12272,-0.10761 -0.0117,0.01 -0.0376,0.0142 -0.035,0.0291 0.004,0.0235 0.0335,0.0341 0.049,0.0522 0.021,0.0247 0.044,0.0483 0.0604,0.0764 0.035,0.0599 0.0298,0.12423 0.0213,0.18912 -0.0179,0.0778 -0.0408,0.17388 -0.12443,0.20273 -0.0304,10e-4 -0.06,0.005 -0.0905,0.002 -0.0195,-0.002 -0.0823,-0.0119 0.007,0.002 -0.5387,-0.15754 -0.76148,0.60431 -0.22278,0.76184 v 0 c 0.0687,0.0156 0.13493,0.0305 0.20579,0.0299 0.19738,-2.6e-4 0.3872,-0.0392 0.55659,-0.14641 0.29001,-0.21872 0.44666,-0.51482 0.46171,-0.882 -0.006,-0.0776 -0.003,-0.1564 -0.0185,-0.23273 -0.0641,-0.32323 -0.29454,-0.6535 -0.60325,-0.7895 -0.10053,-0.0443 -0.21185,-0.0582 -0.31778,-0.0873 -0.0897,0.0153 -0.1825,0.0182 -0.26921,0.046 -0.24225,0.0775 -0.41094,0.30086 -0.52138,0.51647 -0.23849,0.50808 0.48004,0.84535 0.71853,0.33727 z" /> - <path - style="fill:#f0c03b;fill-opacity:1;stroke-width:10.5833;stroke-linejoin:bevel" id="path91" - d="m 152.2881,177.69871 c 0.0368,5.3e-4 0.0735,7.9e-4 0.11028,0.001 0.11883,10e-4 0.23766,0.002 0.35649,0.003 0.0938,7.9e-4 0.11844,-0.008 0.078,0.004 0.5527,0.0977 0.69084,-0.68396 0.13814,-0.78164 v 0 c -0.22266,-0.029 -0.44772,-0.0138 -0.67189,-0.0209 -0.56121,-0.008 -0.5722,0.7859 -0.011,0.79367 z" /> + d="m 152.2881,175.45662 c 0.0368,5.3e-4 0.0735,7.9e-4 0.11028,0.001 0.11883,0.001 0.23766,0.002 0.35649,0.003 0.0938,7.9e-4 0.11844,-0.008 0.078,0.004 0.5527,0.0977 0.69084,-0.68396 0.13814,-0.78164 v 0 c -0.22266,-0.029 -0.44772,-0.0138 -0.67189,-0.0209 -0.56121,-0.008 -0.5722,0.7859 -0.011,0.79367 z" /> <path style="fill:#f0c03b;fill-opacity:1;stroke-width:10.5833;stroke-linejoin:bevel" id="path92" - d="m 154.36864,177.1562 c 0.0722,-0.21667 0.16845,-0.4247 0.26973,-0.6291 0.076,-0.15333 0.0392,-0.10883 0.10443,-0.1776 -0.19793,0.0192 -0.39586,0.0383 -0.59378,0.0575 -0.022,-0.0164 -0.0563,-0.075 -0.0659,-0.0493 -0.0252,0.0677 0.003,0.1445 0.004,0.21676 0.005,0.32675 0.017,0.65166 0.0358,0.97793 0.0266,0.30078 0.0331,0.60691 0.0951,0.90357 0.12576,0.547 0.89933,0.36916 0.77358,-0.17784 v 0 c 0.002,0.0117 0.0132,0.0958 -0.007,-0.0423 -0.005,-0.0343 -0.0102,-0.0686 -0.0144,-0.10299 -0.0254,-0.20801 -0.0418,-0.41709 -0.0546,-0.62621 -0.0145,-0.2499 -0.0243,-0.38597 -0.0309,-0.63841 -8e-4,-0.0302 -0.005,-0.55414 -0.012,-0.59842 -0.0205,-0.12109 0.009,-0.29885 -0.1,-0.35459 -0.19874,-0.10125 -0.44542,-0.0245 -0.66813,-0.0367 -0.22182,0.32097 -0.36781,0.69092 -0.49774,1.05711 -0.156,0.53915 0.60648,0.75977 0.76248,0.22061 z" /> + d="m 154.36864,174.91411 c 0.0722,-0.21667 0.16845,-0.4247 0.26973,-0.6291 0.076,-0.15333 0.0392,-0.10883 0.10443,-0.1776 -0.19793,0.0192 -0.39586,0.0383 -0.59378,0.0575 -0.022,-0.0164 -0.0563,-0.075 -0.0659,-0.0493 -0.0252,0.0677 0.003,0.1445 0.004,0.21676 0.005,0.32675 0.017,0.65166 0.0358,0.97793 0.0266,0.30078 0.0331,0.60691 0.0951,0.90357 0.12576,0.547 0.89933,0.36916 0.77358,-0.17784 v 0 c 0.002,0.0117 0.0132,0.0958 -0.007,-0.0423 -0.005,-0.0343 -0.0102,-0.0686 -0.0144,-0.10299 -0.0254,-0.20801 -0.0418,-0.41709 -0.0546,-0.62621 -0.0145,-0.2499 -0.0243,-0.38597 -0.0309,-0.63841 -8e-4,-0.0302 -0.005,-0.55414 -0.012,-0.59842 -0.0205,-0.12109 0.009,-0.29885 -0.1,-0.35459 -0.19874,-0.10125 -0.44542,-0.0245 -0.66813,-0.0367 -0.22182,0.32097 -0.36781,0.69092 -0.49774,1.05711 -0.156,0.53915 0.60648,0.75977 0.76248,0.22061 z" /> <path style="fill:#000000;fill-opacity:1;stroke-width:10.5833;stroke-linejoin:bevel" id="path93" @@ -336,5 +328,14 @@ id="path6-6-2" d="m 131.32396,170.11952 c -0.0495,-1.02382 -0.1242,-2.04482 -0.12363,-3.07056 0.009,-0.19939 -0.0397,-0.58084 0.0658,-0.84173 0.0321,-0.0794 0.0784,-0.191 0.16375,-0.19808 0.076,-0.006 0.12863,0.0877 0.17284,0.14989 0.0286,0.0401 0.0523,0.088 0.0549,0.13726 0.002,0.0303 -0.011,0.0605 -0.0248,0.0875 -0.0833,0.1629 -0.19731,0.30675 -0.34039,0.43055 -0.3762,0.2477 -0.62591,0.082 -0.11385,0.83208 0.0225,0.0329 0.32234,0.004 0.36225,0.006 0.0553,0.003 0.11499,0.0122 0.16691,0.0315 0.15058,0.0558 0.42704,0.16769 0.41155,0.23178 -0.0257,0.10651 -0.37131,0.16145 -0.58179,0.18471 -0.1084,0.007 -0.0582,0.004 -0.1507,0.007 -0.5609,0.0203 -0.53216,0.81355 0.0287,0.79323 v 0 c 0.13074,-0.005 0.0583,-0.001 0.21722,-0.0126 0.48896,-0.0624 0.92693,-0.22848 1.21385,-0.65483 0.0326,-0.11881 0.0978,-0.23323 0.0978,-0.35643 -1e-5,-0.61164 -0.68987,-0.88231 -1.17411,-1.0006 -0.0813,-0.006 -0.16252,-0.0152 -0.24405,-0.0178 -0.0585,-0.002 -0.17866,-0.0542 -0.17545,0.004 0.014,0.25315 0.0885,0.81922 0.48121,0.53382 0.17097,-0.16439 0.2888,-0.25476 0.4058,-0.47462 0.21124,-0.39694 0.23435,-0.85119 -0.0571,-1.22273 -0.0881,-0.11237 -0.22516,-0.17575 -0.33773,-0.26362 -0.37439,-0.11522 -0.5309,-0.23769 -0.93264,-0.0265 -0.53255,0.27995 -0.47488,1.15125 -0.50375,1.63974 1.9e-4,1.00081 0.0586,1.99837 0.1271,2.99661 -0.0524,0.55882 0.73789,0.63291 0.79028,0.0741 z" sodipodi:nodetypes="ccaaaaacccssccssccsccccsccscccc" /> + <path + style="fill:#f0c03b;fill-opacity:1;stroke-width:10.5833;stroke-linejoin:bevel" + id="path9" + d="m 150.66393,174.54613 c -0.006,-0.0655 -0.002,-0.13456 -0.0205,-0.19832 -0.0115,-0.0387 -0.0269,-0.0762 -0.0404,-0.11431 -0.24499,0.0198 -0.35562,0.17065 -0.59368,0.23195 -0.0177,0.005 -0.1903,0.10898 -0.1906,0.11177 -0.0334,0.30784 -0.0396,0.61576 -0.0469,0.92509 -0.008,0.70937 0.0286,1.41932 0.003,2.12849 -0.004,0.032 -0.007,0.0641 -0.0109,0.0961 -0.0581,0.55825 0.73137,0.64044 0.78948,0.0822 v 0 c 0.005,-0.05 0.01,-0.1 0.0147,-0.14998 0.0248,-0.71255 -0.008,-1.42568 -0.002,-2.13843 0.006,-0.27878 0.005,-0.38819 0.0234,-0.65433 0.004,-0.0634 0.011,-0.1267 0.0176,-0.18995 0.004,-0.0373 0.0114,-0.0742 0.0136,-0.11158 1.5e-4,-0.003 -0.005,0.002 -0.007,0.003 -0.24006,0.006 -0.48013,0.0129 -0.72019,0.0193 -0.01,-0.0303 -0.0192,-0.0606 -0.0287,-0.0909 0.002,0.004 0.003,0.005 0.003,0.0109 0.003,0.0321 0.003,0.0645 0.005,0.0967 0.0408,0.55978 0.83245,0.50207 0.79165,-0.0577 z" + sodipodi:nodetypes="ccccccccsscccccccccccc" /> + <path + style="fill:#f0c03b;fill-opacity:1;stroke-width:10.5833;stroke-linejoin:bevel" + id="path10" + d="m 150.47746,174.8483 c -0.007,0.0239 -0.0279,0.0394 -0.0387,0.0608 -0.002,0.003 0.006,-0.002 0.009,-0.004 0.01,-0.006 0.0191,-0.0123 0.029,-0.018 0.0412,-0.0239 0.0365,-0.0204 0.0797,-0.0396 0.0663,-0.0174 0.14088,-0.0385 0.20965,-0.0202 0.0132,0.004 0.0453,0.0297 0.0546,0.0368 0.0258,0.0265 0.0743,0.0765 0.0354,0.11156 -0.002,0.002 -0.004,0.005 -0.007,0.004 -0.002,-0.002 -0.007,-0.008 -0.0108,-0.008 -0.0156,-8e-4 -0.0559,0.0344 -0.063,0.0393 -0.0915,0.058 -0.1922,0.0979 -0.29654,0.12541 -0.0215,0.005 -0.0264,0.006 -0.0458,0.01 -0.003,5.3e-4 -0.0117,0.002 -0.009,0.001 0.0315,-0.006 0.0243,-0.003 0.0731,-0.004 -0.56001,-0.0376 -0.61318,0.75437 -0.0532,0.79197 v 0 c 0.0839,-2.7e-4 0.16515,-0.01 0.24591,-0.0341 0.20001,-0.0564 0.39276,-0.13639 0.56342,-0.25694 0.0309,-0.0241 0.0631,-0.0464 0.0926,-0.0721 0.18323,-0.15981 0.2857,-0.35218 0.30334,-0.59598 -0.006,-0.13547 0.002,-0.18015 -0.0439,-0.31231 -0.0571,-0.16549 -0.16181,-0.31041 -0.29843,-0.41957 -0.0473,-0.0378 -0.10098,-0.0668 -0.15147,-0.10024 -0.0578,-0.0242 -0.11351,-0.0544 -0.1735,-0.0726 -0.25573,-0.0777 -0.52373,-0.0387 -0.76738,0.0588 -0.17157,0.0841 -0.34482,0.18147 -0.44272,0.35294 -0.2578,0.49856 0.44727,0.86314 0.70506,0.36458 z" /> </g> </svg>
diff --git a/images/lickorish-gens.svg b/images/lickorish-gens.svg @@ -23,16 +23,17 @@ inkscape:pagecheckerboard="1" inkscape:deskcolor="#d1d1d1" inkscape:document-units="mm" - showguides="false" - inkscape:zoom="0.70710678" - inkscape:cx="399.51533" - inkscape:cy="42.426407" + showguides="true" + inkscape:zoom="2" + inkscape:cx="477.5" + inkscape:cy="95.5" inkscape:window-width="1358" inkscape:window-height="728" inkscape:window-x="4" inkscape:window-y="36" inkscape:window-maximized="1" - inkscape:current-layer="layer1"> + inkscape:current-layer="layer1" + showgrid="false"> <sodipodi:guide position="97.054695,18.998846" orientation="0,-1" @@ -265,23 +266,16 @@ <path style="fill:#f0c03b;fill-opacity:1;stroke-width:10.5833;stroke-linejoin:bevel" id="path87" - d="m 145.99197,170.62155 c -0.0154,0.46245 -0.025,0.92504 -0.0324,1.38768 -0.004,0.25842 -0.007,0.51685 -0.009,0.77529 -0.002,0.31367 -0.0133,0.17892 0.7942,0.0845 0.0391,-0.005 0.0133,-0.0776 0.0199,-0.11639 0.0631,-0.35486 0.13788,-0.7139 0.32404,-1.0266 0.0169,-0.018 0.0317,-0.0383 0.0507,-0.054 0.0101,-0.008 0.0222,-0.0162 0.0352,-0.017 0.005,-2.7e-4 0.009,0.0116 0.005,0.0145 -0.0689,0.0483 -0.14623,0.0163 -0.1624,0.0741 0.14757,0.55425 0.0937,1.14167 0.0695,1.70835 -0.0262,0.71116 -0.10631,1.4495 0.0742,2.14932 0.033,0.12788 0.0864,0.2496 0.12966,0.3744 0.20445,0.44055 0.54085,0.86479 1.09519,0.62575 0.22675,-0.0978 0.34934,-0.30753 0.48604,-0.4925 0.10808,-0.1869 0.23793,-0.37539 0.28902,-0.58933 0.0143,-0.0598 0.0157,-0.12198 0.0236,-0.18298 0.0244,-0.56073 -0.76864,-0.59519 -0.79301,-0.0345 v 0 c 0.006,-0.0119 0.019,-0.0487 0.0171,-0.0356 -0.0204,0.14066 -0.12444,0.26076 -0.18441,0.38359 -0.0325,0.0497 -0.12802,0.16383 -0.0995,0.20174 0.002,0.002 0.22865,0.11101 0.10029,0.10261 -0.0869,-0.006 -0.18237,-0.23923 -0.18823,-0.2497 -0.0314,-0.0807 -0.0716,-0.15853 -0.0942,-0.24214 -0.17368,-0.64186 -0.0812,-1.32446 -0.0627,-1.97701 0.0246,-0.58436 0.0978,-1.3914 -0.10353,-1.94666 -0.0355,-0.098 -0.0885,-0.18868 -0.13272,-0.28302 -0.1684,-0.18241 -0.21616,-0.29268 -0.48321,-0.33979 -0.33351,-0.0588 -0.58676,0.17062 -0.76157,0.41749 -0.22116,0.39947 -0.34649,0.83482 -0.41654,1.28553 -0.007,0.0435 -0.0647,0.11844 -0.0222,0.13064 0.80765,0.23211 0.7827,0.33742 0.78427,0.0402 0.002,-0.25599 0.005,-0.51198 0.009,-0.76796 0.007,-0.46164 0.0171,-0.92323 0.0324,-1.38469 0.0111,-0.56116 -0.78245,-0.57691 -0.79359,-0.0157 z" /> - <path - style="fill:#f0c03b;fill-opacity:1;stroke-width:10.5833;stroke-linejoin:bevel" - id="path89" - d="m 149.77933,175.53002 c -0.0142,0.47726 -0.0234,0.95461 -0.0306,1.43202 -0.01,0.34661 0.005,0.69297 0.0126,1.03937 -4e-5,-0.0123 -8e-5,-0.0247 -1.2e-4,-0.037 -0.0797,0.55557 0.70595,0.66834 0.7857,0.11277 v 0 c 0.003,-0.0311 0.005,-0.0622 0.008,-0.0933 -0.008,-0.33654 -0.0215,-0.67303 -0.0125,-1.00978 0.007,-0.47626 0.0164,-0.95247 0.0306,-1.42857 0.011,-0.56116 -0.78266,-0.57664 -0.7936,-0.0155 z" /> - <path - style="fill:#f0c03b;fill-opacity:1;stroke-width:10.5833;stroke-linejoin:bevel" - id="path90" - d="m 150.63956,177.44402 c -0.0149,-0.0561 0.19266,-0.19947 0.12272,-0.10761 -0.0117,0.01 -0.0376,0.0142 -0.035,0.0291 0.004,0.0235 0.0335,0.0341 0.049,0.0522 0.021,0.0247 0.044,0.0483 0.0604,0.0764 0.035,0.0599 0.0298,0.12423 0.0213,0.18912 -0.0179,0.0778 -0.0408,0.17388 -0.12443,0.20273 -0.0304,10e-4 -0.06,0.005 -0.0905,0.002 -0.0195,-0.002 -0.0823,-0.0119 0.007,0.002 -0.5387,-0.15754 -0.76148,0.60431 -0.22278,0.76184 v 0 c 0.0687,0.0156 0.13493,0.0305 0.20579,0.0299 0.19738,-2.6e-4 0.3872,-0.0392 0.55659,-0.14641 0.29001,-0.21872 0.44666,-0.51482 0.46171,-0.882 -0.006,-0.0776 -0.003,-0.1564 -0.0185,-0.23273 -0.0641,-0.32323 -0.29454,-0.6535 -0.60325,-0.7895 -0.10053,-0.0443 -0.21185,-0.0582 -0.31778,-0.0873 -0.0897,0.0153 -0.1825,0.0182 -0.26921,0.046 -0.24225,0.0775 -0.41094,0.30086 -0.52138,0.51647 -0.23849,0.50808 0.48004,0.84535 0.71853,0.33727 z" /> + d="m 145.99197,170.62155 c -0.0154,0.46245 -0.025,0.92504 -0.0324,1.38768 -0.004,0.25842 -0.007,0.51685 -0.009,0.77529 -0.002,0.31367 -0.0133,0.17892 0.7942,0.0845 0.0391,-0.005 0.0133,-0.0776 0.0199,-0.11639 0.0631,-0.35486 0.13788,-0.7139 0.32404,-1.0266 0.0169,-0.018 0.0317,-0.0383 0.0507,-0.054 0.0101,-0.008 0.0222,-0.0162 0.0352,-0.017 0.005,-2.7e-4 0.009,0.0116 0.005,0.0145 -0.0689,0.0483 -0.14623,0.0163 -0.1624,0.0741 0.14757,0.55425 0.0937,1.14167 0.0695,1.70835 -0.0262,0.71116 -0.10631,1.4495 0.0742,2.14932 0.033,0.12788 0.0864,0.2496 0.12966,0.3744 0.20445,0.44055 0.54085,0.86479 1.09519,0.62575 0.22675,-0.0978 0.34934,-0.30753 0.48604,-0.4925 0.10808,-0.1869 0.23793,-0.37539 0.28902,-0.58933 0.0143,-0.0598 0.0157,-0.12198 0.0236,-0.18298 0.0244,-0.56073 -0.75498,-0.56322 -0.79301,-0.0345 -0.005,0.0661 -0.10673,0.24324 -0.16731,0.34799 -0.0464,0.0829 -0.14084,0.13788 -0.18744,0.0547 -0.0314,-0.0807 -0.0716,-0.15853 -0.0942,-0.24214 -0.17368,-0.64186 -0.0812,-1.32446 -0.0627,-1.97701 0.0246,-0.58436 0.0978,-1.3914 -0.10353,-1.94666 -0.0355,-0.098 -0.0885,-0.18868 -0.13272,-0.28302 -0.1684,-0.18241 -0.21616,-0.29268 -0.48321,-0.33979 -0.33351,-0.0588 -0.58676,0.17062 -0.76157,0.41749 -0.22116,0.39947 -0.34649,0.83482 -0.41654,1.28553 -0.007,0.0435 -0.0647,0.11844 -0.0222,0.13064 0.80765,0.23211 0.7827,0.33742 0.78427,0.0402 0.002,-0.25599 0.005,-0.51198 0.009,-0.76796 0.007,-0.46164 0.0171,-0.92323 0.0324,-1.38469 0.0111,-0.56116 -0.78245,-0.57691 -0.79359,-0.0157 z" + sodipodi:nodetypes="cccccccccccscscscsccccccccccccccc" /> <path style="fill:#f0c03b;fill-opacity:1;stroke-width:10.5833;stroke-linejoin:bevel" id="path91" - d="m 152.2881,177.69871 c 0.0368,5.3e-4 0.0735,7.9e-4 0.11028,0.001 0.11883,10e-4 0.23766,0.002 0.35649,0.003 0.0938,7.9e-4 0.11844,-0.008 0.078,0.004 0.5527,0.0977 0.69084,-0.68396 0.13814,-0.78164 v 0 c -0.22266,-0.029 -0.44772,-0.0138 -0.67189,-0.0209 -0.56121,-0.008 -0.5722,0.7859 -0.011,0.79367 z" /> + d="m 152.48897,175.25568 c 0.0368,5.3e-4 0.0735,7.9e-4 0.11028,10e-4 0.11883,0.001 0.23766,0.002 0.35649,0.003 0.0938,7.9e-4 0.11844,-0.008 0.078,0.004 0.5527,0.0977 0.69084,-0.68396 0.13814,-0.78164 v 0 c -0.22266,-0.029 -0.44772,-0.0138 -0.67189,-0.0209 -0.56121,-0.008 -0.5722,0.7859 -0.011,0.79367 z" /> <path style="fill:#f0c03b;fill-opacity:1;stroke-width:10.5833;stroke-linejoin:bevel" id="path92" - d="m 154.36864,177.1562 c 0.0722,-0.21667 0.16845,-0.4247 0.26973,-0.6291 0.076,-0.15333 0.0392,-0.10883 0.10443,-0.1776 -0.19793,0.0192 -0.39586,0.0383 -0.59378,0.0575 -0.022,-0.0164 -0.0563,-0.075 -0.0659,-0.0493 -0.0252,0.0677 0.003,0.1445 0.004,0.21676 0.005,0.32675 0.017,0.65166 0.0358,0.97793 0.0266,0.30078 0.0331,0.60691 0.0951,0.90357 0.12576,0.547 0.89933,0.36916 0.77358,-0.17784 v 0 c 0.002,0.0117 0.0132,0.0958 -0.007,-0.0423 -0.005,-0.0343 -0.0102,-0.0686 -0.0144,-0.10299 -0.0254,-0.20801 -0.0418,-0.41709 -0.0546,-0.62621 -0.0145,-0.2499 -0.0243,-0.38597 -0.0309,-0.63841 -8e-4,-0.0302 -0.005,-0.55414 -0.012,-0.59842 -0.0205,-0.12109 0.009,-0.29885 -0.1,-0.35459 -0.19874,-0.10125 -0.44542,-0.0245 -0.66813,-0.0367 -0.22182,0.32097 -0.36781,0.69092 -0.49774,1.05711 -0.156,0.53915 0.60648,0.75977 0.76248,0.22061 z" /> + d="m 154.56951,174.71317 c 0.0722,-0.21667 0.16845,-0.4247 0.26973,-0.6291 0.076,-0.15333 0.0392,-0.10883 0.10443,-0.1776 -0.19793,0.0192 -0.39586,0.0383 -0.59378,0.0575 -0.022,-0.0164 -0.0563,-0.075 -0.0659,-0.0493 -0.0252,0.0677 0.003,0.1445 0.004,0.21676 0.005,0.32675 0.017,0.65166 0.0358,0.97793 0.0266,0.30078 0.0331,0.60691 0.0951,0.90357 0.12576,0.547 0.89933,0.36916 0.77358,-0.17784 v 0 c 0.002,0.0117 0.0132,0.0958 -0.007,-0.0423 -0.005,-0.0343 -0.0102,-0.0686 -0.0144,-0.10299 -0.0254,-0.20801 -0.0418,-0.41709 -0.0546,-0.62621 -0.0145,-0.2499 -0.0243,-0.38597 -0.0309,-0.63841 -8e-4,-0.0302 -0.005,-0.55414 -0.012,-0.59842 -0.0205,-0.12109 0.009,-0.29885 -0.1,-0.35459 -0.19874,-0.10125 -0.44542,-0.0245 -0.66813,-0.0367 -0.22182,0.32097 -0.36781,0.69092 -0.49774,1.05711 -0.156,0.53915 0.60648,0.75977 0.76248,0.22061 z" /> <path style="fill:#000000;fill-opacity:1;stroke-width:10.5833;stroke-linejoin:bevel" id="path93" @@ -323,5 +317,14 @@ style="fill:#568bb5;fill-opacity:1;stroke-width:10.5833;stroke-linejoin:bevel" id="path8" d="m 150.6988,168.43819 c -0.0623,-0.64564 -0.0972,-1.29227 -0.0988,-1.94085 0.007,-0.22804 0.0316,-1.07852 0.26516,-1.2288 0.0189,-0.0122 0.0269,0.0365 0.0435,0.0516 0.0179,0.0163 0.0395,0.0281 0.0592,0.0421 0.0459,0.0335 0.22465,0.12116 0.16095,0.21034 -0.012,0.0168 -0.0434,0.006 -0.0591,0.019 -0.0282,0.0244 -0.0432,0.0608 -0.0667,0.0897 -0.0344,0.0424 -0.0712,0.0828 -0.10675,0.12425 -0.17179,0.18536 -0.39671,0.29827 -0.60432,0.43464 0.1291,0.23184 0.22589,0.4849 0.38731,0.69553 0.0261,0.0341 0.0839,-0.0188 0.12638,-0.0252 0.22326,-0.0337 0.42417,-0.0225 0.63768,0.0489 0.0706,0.0354 0.3254,0.16326 0.33545,0.26098 0.002,0.0146 -0.0289,-0.0124 -0.036,-0.0252 -0.0164,-0.0295 -0.021,-0.0641 -0.0316,-0.0962 -0.002,-0.0168 0.008,-0.0409 -0.006,-0.0505 -0.007,-0.005 -0.10224,0.0519 -0.13611,0.0673 -0.16196,0.0738 -0.3324,0.12024 -0.50357,0.16656 -0.20248,0.0482 -0.4066,0.092 -0.61328,0.11797 -0.006,5.3e-4 -0.0121,0.001 -0.0182,0.002 -0.55821,0.0585 -0.47548,0.84793 0.0827,0.78943 v 0 c 0.0119,-0.001 0.0239,-0.003 0.0358,-0.004 0.24367,-0.0314 0.4847,-0.0813 0.72318,-0.13997 0.37043,-0.10299 0.83754,-0.22001 1.08199,-0.54742 0.0563,-0.0754 0.0881,-0.16641 0.13219,-0.24962 0.003,-0.0997 0.0254,-0.20086 0.008,-0.29902 -0.0698,-0.3921 -0.51439,-0.61296 -0.83356,-0.75459 -0.38487,-0.10311 -0.62162,-0.1279 -1.01737,-0.0645 -0.0552,0.009 -0.17087,-0.0216 -0.16412,0.0339 0.0311,0.25569 0.17127,0.48584 0.25691,0.72876 0.27901,-0.16655 0.57028,-0.35729 0.78986,-0.61179 0.15027,-0.21226 0.29662,-0.36733 0.33193,-0.64234 0.0545,-0.42447 -0.14336,-0.77683 -0.52431,-0.96205 -0.11281,-0.0549 -0.23978,-0.0738 -0.35967,-0.11062 -0.13772,0.012 -0.28339,-0.0115 -0.41315,0.0361 -0.74496,0.27361 -0.72176,1.25394 -0.7586,1.89517 0.003,0.65775 0.0337,1.31401 0.0993,1.96869 0.0215,0.56085 0.81466,0.53045 0.79317,-0.0304 z" /> + <path + style="fill:#f0c03b;fill-opacity:1;stroke-width:10.5833;stroke-linejoin:bevel" + id="path9" + d="m 150.43683,174.60402 c -0.006,-0.0655 -0.002,-0.13456 -0.0205,-0.19832 -0.0115,-0.0387 -0.0269,-0.0762 -0.0404,-0.11431 -0.24499,0.0198 -0.35562,0.17065 -0.59368,0.23195 -0.0177,0.005 -0.1903,0.10898 -0.1906,0.11177 -0.0334,0.30784 -0.0396,0.61576 -0.0469,0.92509 -0.008,0.70937 0.0286,1.41932 0.003,2.12849 -0.004,0.032 -0.007,0.0641 -0.0109,0.0961 -0.0581,0.55825 0.73137,0.64044 0.78948,0.0822 v 0 c 0.005,-0.05 0.01,-0.1 0.0147,-0.14998 0.0248,-0.71255 -0.008,-1.42568 -0.002,-2.13843 0.006,-0.27878 0.005,-0.38819 0.0234,-0.65433 0.004,-0.0634 0.011,-0.1267 0.0176,-0.18995 0.004,-0.0373 0.0114,-0.0742 0.0136,-0.11158 1.5e-4,-0.003 -0.005,0.002 -0.007,0.003 -0.24006,0.006 -0.48013,0.0129 -0.72019,0.0193 -0.01,-0.0303 -0.0192,-0.0606 -0.0287,-0.0909 0.002,0.004 0.003,0.005 0.003,0.0109 0.003,0.0321 0.003,0.0645 0.005,0.0967 0.0408,0.55978 0.83245,0.50207 0.79165,-0.0577 z" + sodipodi:nodetypes="ccccccccsscccccccccccc" /> + <path + style="fill:#f0c03b;fill-opacity:1;stroke-width:10.5833;stroke-linejoin:bevel" + id="path10" + d="m 150.25036,174.90619 c -0.007,0.0239 -0.0279,0.0394 -0.0387,0.0608 -0.002,0.003 0.006,-0.002 0.009,-0.004 0.01,-0.006 0.0191,-0.0123 0.029,-0.018 0.0412,-0.0239 0.0365,-0.0204 0.0797,-0.0396 0.0663,-0.0174 0.14088,-0.0385 0.20965,-0.0202 0.0132,0.004 0.0453,0.0297 0.0546,0.0368 0.0258,0.0265 0.0743,0.0765 0.0354,0.11156 -0.002,0.002 -0.004,0.005 -0.007,0.004 -0.002,-0.002 -0.007,-0.008 -0.0108,-0.008 -0.0156,-8e-4 -0.0559,0.0344 -0.063,0.0393 -0.0915,0.058 -0.1922,0.0979 -0.29654,0.12541 -0.0215,0.005 -0.0264,0.006 -0.0458,0.01 -0.003,5.3e-4 -0.0117,0.002 -0.009,0.001 0.0315,-0.006 0.0243,-0.003 0.0731,-0.004 -0.56001,-0.0376 -0.61318,0.75437 -0.0532,0.79197 v 0 c 0.0839,-2.7e-4 0.16515,-0.01 0.24591,-0.0341 0.20001,-0.0564 0.39276,-0.13639 0.56342,-0.25694 0.0309,-0.0241 0.0631,-0.0464 0.0926,-0.0721 0.18323,-0.15981 0.2857,-0.35218 0.30334,-0.59598 -0.006,-0.13547 0.002,-0.18015 -0.0439,-0.31231 -0.0571,-0.16549 -0.16181,-0.31041 -0.29843,-0.41957 -0.0473,-0.0378 -0.10098,-0.0668 -0.15147,-0.10024 -0.0578,-0.0242 -0.11351,-0.0544 -0.1735,-0.0726 -0.25573,-0.0777 -0.52373,-0.0387 -0.76738,0.0588 -0.17157,0.0841 -0.34482,0.18147 -0.44272,0.35294 -0.2578,0.49856 0.44727,0.86314 0.70506,0.36458 z" /> </g> </svg>
diff --git a/sections/introduction.tex b/sections/introduction.tex @@ -46,18 +46,18 @@ geometry of the curves in \(S\) and their intersections. For example\dots \end{lemma} \begin{proof} - Let \(S = S_{g, r}^b\) and consider the surface \(S_{\alpha \alpha'}\) + Let \(S = S_{g, r}^p\) and consider the surface \(S_{\alpha \alpha'}\) obtained by cutting \(S\) across \(\alpha\) and \(\alpha'\), as in Figure~\ref{fig:change-of-coordinates}. Since \(\alpha\) and \(\alpha'\) are nonseparating, this surface has genus \(g - 1\) and one additional boundary component \(\delta \subset \partial S_{\alpha \beta}\), so \(S_{\alpha \beta} - \cong S_{g-1,r}^{b+1}\). The boundary component \(\delta\) is naturally + \cong S_{g-1,r}^{p+1}\). The boundary component \(\delta\) is naturally subdived into the four arcs in Figure~\ref{fig:change-of-coordinates}, each corresponding to one of the curves \(\alpha\) and \(\alpha'\) in \(S\). By identifying the pairs of arcs corresponding to the same curve we obtain the surface \(\mfrac{S_{\alpha \beta}}{\sim} \cong S\). - Similarly, \(S_{\beta \beta'} \cong S_{g-1, r}^{b+1}\) also has an additional + Similarly, \(S_{\beta \beta'} \cong S_{g-1, r}^{p+1}\) also has an additional boundary component \(\delta' \subset \partial S_{\beta \beta'}\) subdivided into four arcs. Now by the classification of surfaces we can find an orientation-preserving homemorphism \(\tilde\phi : S_{\alpha \alpha'} \isoto @@ -72,10 +72,10 @@ geometry of the curves in \(S\) and their intersections. For example\dots \begin{figure}[ht] \centering \includegraphics[width=.8\linewidth]{images/change-of-coords-cut.eps} - \caption{By cutting $S_{g, r}^b$ across $\alpha$ we obtain $S_{g-1, - r}^{b+2}$, where $\alpha'$ deterimines a yellow arc joining the two - additional boundary components. Now by cutting $S_{g-1, r}^{b+2}$ across this - arc we obtain $S_{g-1,r}^b$, with the added boundary component subdivided + \caption{By cutting $S_{g, r}^p$ across $\alpha$ we obtain $S_{g-1, + r}^{p+2}$, where $\alpha'$ deterimines a yellow arc joining the two + additional boundary components. Now by cutting $S_{g-1, r}^{p+2}$ across this + arc we obtain $S_{g-1,r}^p$, with the added boundary component subdivided into the four arcs corresponding to $\alpha$ and $\alpha'$.} \label{fig:change-of-coordinates} \end{figure}
diff --git a/sections/presentation.tex b/sections/presentation.tex @@ -38,21 +38,21 @@ interesting relations between the corresponding Dehn twists in \(\Mod(S)\). For example\dots \begin{proposition}\label{thm:trivial-abelianization} - The Abelianization \(\Mod(S_g^b)^\ab = \mfrac{\Mod(S_g^b)}{[\Mod(S_g), - \Mod(S_g)]}\) is cyclic. Moreover, if \(g \ge 3\) then \(\Mod(S_g^b)^\ab = + The Abelianization \(\Mod(S_g^p)^\ab = \mfrac{\Mod(S_g^p)}{[\Mod(S_g), + \Mod(S_g)]}\) is cyclic. Moreover, if \(g \ge 3\) then \(\Mod(S_g^p)^\ab = 1\). In other words, \(\Mod(S_g)\) is a perfect group for \(g \ge 3\). \end{proposition} \begin{proof} - By Theorem~\ref{thm:lickorish-gens}, \(\Mod(S_g^b)^\ab\) is generated by the + By Theorem~\ref{thm:lickorish-gens}, \(\Mod(S_g^p)^\ab\) is generated by the image of the Lickrish generators, which are all conjugate and thus represent the same class in the Abelianization. In fact, any nonseparating \(\alpha - \subset S_g^b\) is conjugate to the Lickorish generators too, so - \(\Mod(S_g^b)^\ab = \langle [\alpha] \rangle\). + \subset S_g^p\) is conjugate to the Lickorish generators too, so + \(\Mod(S_g^p)^\ab = \langle [\alpha] \rangle\). - Now for \(g \ge 3\) we can embed \(S_0^4\) in \(S_g^b\) in such a way that + Now for \(g \ge 3\) we can embed \(S_0^4\) in \(S_g^p\) in such a way that all the corresponding curves \(\alpha, \beta, \gamma, \delta_1, \ldots, - \delta_4 \subset S_g^b\) are nonseparating, as shown in + \delta_4 \subset S_g^p\) are nonseparating, as shown in Figure~\ref{fig:latern-relation-trivial-abelianization}. The lantern relation (\ref{eq:latern-relation}) then becomes \[ @@ -62,14 +62,14 @@ example\dots + [\tau_{\delta_3}] + [\tau_{\delta_4}] = 4 \cdot [\tau_\alpha] \] - in \(\Mod(S_g^b)^\ab\). In other words, \([\tau_\alpha] = 0\) and thus - \(\Mod(S_g^b)^\ab = 0\). + in \(\Mod(S_g^p)^\ab\). In other words, \([\tau_\alpha] = 0\) and thus + \(\Mod(S_g^p)^\ab = 0\). \end{proof} \begin{figure}[ht] \centering \includegraphics[width=.5\linewidth]{images/lantern-relation-trivial-abelianization.eps} - \caption{The embedding of $S_0^4$ in $S_g^b$ for $g \ge 3$.} + \caption{The embedding of $S_0^4$ in $S_g^p$ for $g \ge 3$.} \label{fig:latern-relation-trivial-abelianization} \end{figure} @@ -89,14 +89,14 @@ presentations of \(\Mod(S_g)\) for \(g \le 2\) to show the Abelianization is giv for closed surfaces with small genus. In \cite{korkmaz-mccarthy} Korkmaz-McCarthy showed that -eventhough \(\Mod(S_2^b)\) is not perfect, its commutator subgroup is. -In addition, they also show \([\Mod(S_g^b), \Mod(S_g^b)]\) is normaly generated +eventhough \(\Mod(S_2^p)\) is not perfect, its commutator subgroup is. +In addition, they also show \([\Mod(S_g^p), \Mod(S_g^p)]\) is normaly generated by a single mapping class. \begin{proposition}\label{thm:commutator-is-perfect} - The commutator subgroup \(\Mod(S_2^b)' = [\Mod(S_2^b), \Mod(S_2^b)]\) is - perfect -- i.e. \(\Mod(S_2^b)^{(2)} = [\Mod(S_2^b)', \Mod(S_2^b)']\) is the - whole of \(\Mod(S_2^b)'\). + The commutator subgroup \(\Mod(S_2^p)' = [\Mod(S_2^p), \Mod(S_2^p)]\) is + perfect -- i.e. \(\Mod(S_2^p)^{(2)} = [\Mod(S_2^p)', \Mod(S_2^p)']\) is the + whole of \(\Mod(S_2^p)'\). \end{proposition} \begin{proposition}\label{thm:commutator-normal-gen} @@ -217,10 +217,10 @@ we get\dots To get from \(S_{0, n}^1\) to surfaces of genus \(g > 0\) we may consider the \emph{hyperelliptic involution} \(\iota : S_g \isoto S_g\) which rotates \(S_g\) by \(\pi\) around some axis, as shown in -Figure~\ref{fig:hyperelliptic-involution}. Given \(\ell < g\) and \(b = 1, 2\), -we can also embed \(S_\ell^b\) in \(S_g\) in such way that \(\iota\) restricts -to an involution\footnote{This involution does not fix $\partial S_\ell^b$ -point-wise.} \(S_\ell^b \isoto S_\ell^b\). +Figure~\ref{fig:hyperelliptic-involution}. Given \(\ell < g\) and \(p = 1, 2\), +we can also embed \(S_\ell^p\) in \(S_g\) in such way that \(\iota\) restricts +to an involution\footnote{This involution does not fix $\partial S_\ell^p$ +point-wise.} \(S_\ell^p \isoto S_\ell^p\). \begin{figure}[ht] \centering @@ -232,13 +232,13 @@ point-wise.} \(S_\ell^b \isoto S_\ell^b\). It is clear from Figure~\ref{fig:hyperelliptic-involution} that the quotients \(\mfrac{S_\ell^1}{\iota}\) and \(\mfrac{S_\ell^2}{\iota}\) are both disks, with boundary corresponding to the projection of the boundaries of \(S_\ell^1\) -and \(S_\ell^2\), respectively. Given \(b = 1, 2\), the quotient map \(S_\ell^b -\to \mfrac{S_\ell^b}{\iota} \cong \mathbb{D}^2\) is a double cover with \(2\ell + +and \(S_\ell^2\), respectively. Given \(p = 1, 2\), the quotient map \(S_\ell^p +\to \mfrac{S_\ell^p}{\iota} \cong \mathbb{D}^2\) is a double cover with \(2\ell + b\) branch points corresponding to the fixed points of \(\iota\). We may thus -regard \(\mfrac{S_\ell^b}{\iota}\) as the disk \(S_{0, 2\ell + b}^1\) with +regard \(\mfrac{S_\ell^p}{\iota}\) as the disk \(S_{0, 2\ell + b}^1\) with \(2\ell + b\) punctures in its interior, as shown in Figure~\ref{fig:hyperelliptic-covering}. We also draw the curves \(\alpha_1, -\ldots, \alpha_{2\ell} \subset S_\ell^b\) of the Humphreys generators of +\ldots, \alpha_{2\ell} \subset S_\ell^p\) of the Humphreys generators of \(\Mod(S_g)\). Since these curves are invariant under the action of \(\iota\), they descend to arcs \(\bar{\alpha}_1, \ldots, \bar{\alpha}_{2\ell + b} \subset S_{0, 2\ell + b}^1\) joining the punctures of the quotient surface. @@ -262,36 +262,36 @@ the quotient surfaces and their mapping classes, known as \emph{the symmetric mapping clases}. \begin{definition} - Let \(\ell \ge 0\) and \(b = 1, 2\). The \emph{group of symmetric - homeomorphisms of \(S_\ell^b\)} is \(\SHomeo^+(S_\ell^b, \partial S_\ell^b) = - \{\phi \in \Homeo^+(S_\ell^b, \partial S_\ell^b) : [\phi, \iota] = 1\}\). The - \emph{symmetric mapping class group of \(S_\ell^b\)} is the subgroup - \(\SMod(S_\ell^1) = \{ [\phi] \in \Mod(S_\ell^b) : \phi \in - \SHomeo^+(S_\ell^b, \partial S_\ell^b) \}\). + Let \(\ell \ge 0\) and \(p = 1, 2\). The \emph{group of symmetric + homeomorphisms of \(S_\ell^p\)} is \(\SHomeo^+(S_\ell^p, \partial S_\ell^p) = + \{\phi \in \Homeo^+(S_\ell^p, \partial S_\ell^p) : [\phi, \iota] = 1\}\). The + \emph{symmetric mapping class group of \(S_\ell^p\)} is the subgroup + \(\SMod(S_\ell^1) = \{ [\phi] \in \Mod(S_\ell^p) : \phi \in + \SHomeo^+(S_\ell^p, \partial S_\ell^p) \}\). \end{definition} -Fix \(b = 1\) or \(2\). It follows from the universal property of quotients -that any \(\phi \in \SHomeo^+(S_\ell^b, \partial S_\ell^b)\) defines a +Fix \(p = 1\) or \(2\). It follows from the universal property of quotients +that any \(\phi \in \SHomeo^+(S_\ell^p, \partial S_\ell^p)\) defines a homeomorphism \(\bar \phi : S_{0, 2\ell+b}^1 \isoto S_{0, 2\ell+b}^1\). This yeilds a homomorphism of topological groups \begin{align*} - \SHomeo^+(S_\ell^b, \partial S_\ell^b) + \SHomeo^+(S_\ell^p, \partial S_\ell^p) & \to \Homeo^+(S_{0, 2\ell + b}^1, \partial S_{0, 2\ell + b}^1) \\ \phi & \mapsto \bar \phi, \end{align*} which is surjective because any \(\psi \in \Homeo^+(S_{0, 2\ell + b}^1, -\partial S_{0, 2\ell + b}^1)\) lifts to \(S_\ell^b\). +\partial S_{0, 2\ell + b}^1)\) lifts to \(S_\ell^p\). -It is also not hard to see \(\SHomeo^+(S_\ell^b, \partial S_\ell^b) \to +It is also not hard to see \(\SHomeo^+(S_\ell^p, \partial S_\ell^p) \to \Homeo^+(S_{0, 2\ell + b}^1, \partial S_{0, 2\ell + b}^1)\) is injective: the only cadidates for elements of its kernel are \(1\) and \(\iota\), but -\(\iota\) is not an element of \(\SHomeo^+(S_\ell^b, \partial S_\ell^b)\) since -it does not fix \(\partial S_\ell^b\) point-wise. Now since we have a +\(\iota\) is not an element of \(\SHomeo^+(S_\ell^p, \partial S_\ell^p)\) since +it does not fix \(\partial S_\ell^p\) point-wise. Now since we have a continuous bijective homomorphism we find \[ \begin{split} - \pi_0(\SHomeo^+(S_\ell^b, \partial S_\ell^b)) + \pi_0(\SHomeo^+(S_\ell^p, \partial S_\ell^p)) & \cong \pi_0(\Homeo^+(S_{0, 2\ell+b}^1, \partial S_{0, 2\ell+b}^1)) \\ & = \mfrac{\Homeo^+(S_{0,2\ell+b}^1, \partial S_{0, 2\ell+b}^1)}{\simeq} \\ & = \Mod(S_{0, 2\ell+b}^1) \\ @@ -299,27 +299,27 @@ continuous bijective homomorphism we find \end{split} \] -We would like to say \(\pi_0(\SHomeo^+(S_\ell^b, \partial S_\ell^b)) = -\SMod(S_\ell^b)\), but a priori the story looks a little more complicated: -\(\phi, \psi \in \SHomeo^+(S_\ell^b, \partial S_\ell^b)\) define the same class -in \(\SMod(S_\ell^b)\) if they are isotopic, but they may not lie in same -connected component of \(\SHomeo^+(S_\ell^b, \partial S_\ell^b)\) if they are +We would like to say \(\pi_0(\SHomeo^+(S_\ell^p, \partial S_\ell^p)) = +\SMod(S_\ell^p)\), but a priori the story looks a little more complicated: +\(\phi, \psi \in \SHomeo^+(S_\ell^p, \partial S_\ell^p)\) define the same class +in \(\SMod(S_\ell^p)\) if they are isotopic, but they may not lie in same +connected component of \(\SHomeo^+(S_\ell^p, \partial S_\ell^p)\) if they are not isotopic \emph{through symmetric homeomorphisms}. Birman-Hilden \cite{birman-hilden} showed that this is never the case. \begin{theorem}[Birman-Hilden] - If \(\phi, \psi \in \SHomeo^+(S_\ell^b, \partial S_\ell^b)\) are isotopic + If \(\phi, \psi \in \SHomeo^+(S_\ell^p, \partial S_\ell^p)\) are isotopic then \(\phi\) and \(\psi\) are isotopic through symmetric homeomorphisms. In particular, there is an isomorphism \begin{align*} - \SMod(S_\ell^b) & \isoto \Mod(S_{0, 2\ell + b}) \\ + \SMod(S_\ell^p) & \isoto \Mod(S_{0, 2\ell + b}) \\ [\phi] & \mapsto [\bar \phi]. \end{align*} \end{theorem} \begin{example} Using the notation of Figure~\ref{fig:hyperelliptic-covering}, the - Birman-Hilden isomorphism \(\SMod(S_\ell^b) \isoto \Mod(S_{0, 2g + b})\) + Birman-Hilden isomorphism \(\SMod(S_\ell^p) \isoto \Mod(S_{0, 2g + b})\) takes \(\tau_{\alpha_i}\) to the half twist \(h_{\bar{\alpha}_i} \in \Mod(S_{0, 2g + b})\). This can be checked by looking at \(\iota\)-invaratiant anular neighborhoods of the curves \(\alpha_i\) -- @@ -335,7 +335,7 @@ not isotopic \emph{through symmetric homeomorphisms}. Birman-Hilden \(\tau_{\bar\delta_1} = \tau_{\bar\delta_2}\). In light of Example~\ref{ex:push-generators-description}, Example~\ref{ex:braid-group-center} gives us the so called \emph{\(k\)-chain - relations} in \(\SMod(S_\ell^b) \subset \Mod(S_g)\). + relations} in \(\SMod(S_\ell^p) \subset \Mod(S_g)\). \[ \arraycolsep=1.4pt \begin{array}{rlcrll}
diff --git a/sections/representations.tex b/sections/representations.tex @@ -16,8 +16,8 @@ goal of this chapter is providing a concise account of Korkmaz' results, starting by\dots \begin{theorem}[Korkmaz]\label{thm:low-dim-reps-are-trivial} - Let \(S_g^b\) be the surface of genus \(g \ge 1\) and \(b\) boundary - components and \(\rho : \Mod(S_g^b) \to \GL_n(\mathbb{C})\) be a linear + Let \(S_g^p\) be the surface of genus \(g \ge 1\) and \(b\) boundary + components and \(\rho : \Mod(S_g^p) \to \GL_n(\mathbb{C})\) be a linear representation with \(n < 2 g\). Then the image of \(\rho\) is Abelian. In particular, if \(g \ge 3\) then \(\rho\) is trivial. \end{theorem} @@ -28,15 +28,15 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g = 2\). \begin{proposition}\label{thm:low-dim-reps-are-trivial-base-case} - Given \(\rho : \Mod(S_2^b) \to \GL_n(\mathbb{C})\) with \(n \le 3\), the + Given \(\rho : \Mod(S_2^p) \to \GL_n(\mathbb{C})\) with \(n \le 3\), the image of \(\rho\) is Abelian. \end{proposition} \begin{proof}[Sketch of proof] - Given \(\alpha \subset S_2^b\), let \(L_\alpha = \rho(\tau_\alpha)\) and + Given \(\alpha \subset S_2^p\), let \(L_\alpha = \rho(\tau_\alpha)\) and denote by \(E_{\alpha = \lambda} = \{ v \in \mathbb{C}^n : L_\alpha v = \lambda v \}\) its eigenspaces. Let \(\alpha_1, \alpha_2, \beta_1, \beta_2, - \gamma, \eta_1, \ldots, \eta_{b - 1} \subset S_2^b\) be the curves of the + \gamma, \eta_1, \ldots, \eta_{p-1} \subset S_2^p\) be the curves of the Lickorish generators from Theorem~\ref{thm:lickorish-gens}, as shown in Figure~\ref{fig:lickorish-gens-genus-2}. \begin{figure} @@ -46,12 +46,12 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g \label{fig:lickorish-gens-genus-2} \end{figure} - If \(n = 1\) then \(\rho(\Mod(S_2^b)) \subset \GL_1(\mathbb{C}) = + If \(n = 1\) then \(\rho(\Mod(S_2^p)) \subset \GL_1(\mathbb{C}) = \mathbb{C}^\times\) is Abelian. Now if \(n = 2\) or \(3\), by Propositon~\ref{thm:commutator-normal-gen} it suffices to show \(L_{\alpha_1} = L_{\beta_1}\), so that \(\tau_{\alpha_1} \tau_{\beta_1}^{-1} \in \ker - \rho\) and thus \(\Mod(S_2^b)' \subset \ker \rho\) -- i.e. - \(\rho(\Mod(S_2^b))\) is Abelian. Given the braid relation + \rho\) and thus \(\Mod(S_2^p)' \subset \ker \rho\) -- i.e. + \(\rho(\Mod(S_2^p))\) is Abelian. Given the braid relation \begin{equation}\label{eq:braid-rel-induction-basis} L_{\alpha_1} L_{\beta_1} L_{\alpha_1} = L_{\beta_1} L_{\alpha_1} L_{\beta_1}, @@ -138,8 +138,8 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g case. We claim that if \(E_{\alpha_2 = \lambda} = E_{\beta_2 = \lambda}\) then - \(E_{\alpha_2 = \lambda}\) is \(\Mod(S_2^b)\)-invariant. Indeed, by change of - coordinates we can always find \(f, g, h_i \in \Mod(S_2^b)\) with + \(E_{\alpha_2 = \lambda}\) is \(\Mod(S_2^p)\)-invariant. Indeed, by change of + coordinates we can always find \(f, g, h_i \in \Mod(S_2^p)\) with \begin{align*} f \cdot [\alpha_2] & = [\alpha_1] & @@ -184,16 +184,16 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g \end{align*} In other words, \(E_{\alpha_1 = \lambda} = E_{\alpha_2 = \lambda} = E_{\beta_1 = \lambda} = E_{\beta_2 = \lambda} = E_{\gamma = \lambda} = - E_{\eta_1 = \lambda} = \cdots = E_{\eta_{b - 1} = \lambda}\) is invariant + E_{\eta_1 = \lambda} = \cdots = E_{\eta_{p-1} = \lambda}\) is invariant under the action of all Lickorish generators. - Hence \(\rho\) restricts to a subrepresentation \(\bar \rho : \Mod(S_2^b) \to + Hence \(\rho\) restricts to a subrepresentation \(\bar \rho : \Mod(S_2^p) \to \GL(E_{\alpha_2 = \lambda}) = \GL_2(\mathbb{C})\) -- recall \(E_{\alpha_2 = \lambda} = \mathbb{C} e_1 \oplus \mathbb{C} e_2\). By case (2), \(\bar - \rho(f) = 1\) for all \(f \in \Mod(S_2^b)'\), given that \(\bar - \rho(\Mod(S_2^b))\) is Abelian. Thus + \rho(f) = 1\) for all \(f \in \Mod(S_2^p)'\), given that \(\bar + \rho(\Mod(S_2^p))\) is Abelian. Thus \[ - \rho(\Mod(S_2^b)') \subset + \rho(\Mod(S_2^p)') \subset \begin{pmatrix} 1 & 0 & * \\ 0 & 1 & * \\ @@ -202,7 +202,7 @@ by induction on \(g\) and tedious case analysis. We begin by the base case \(g \] lies inside the group of upper triangular matrices, a solvalbe subgroup of \(\GL_3(\mathbb{C})\). Now by Proposition~\ref{thm:commutator-is-perfect} we - get \(\rho(\Mod(S_2^b)') = 1\): any homomorphism from a perfect group to a + get \(\rho(\Mod(S_2^p)') = 1\): any homomorphism from a perfect group to a solvable group is trivial. Finally, if \(E_{\alpha_2 = \lambda} \ne E_{\beta_2 = \lambda}\) and @@ -235,18 +235,18 @@ We are now ready to establish the triviality of low-dimensional representations. \begin{proof}[Proof of Theorem~\ref{thm:low-dim-reps-are-trivial}] - Let \(g \ge 1\), \(b \ge 0\) and fix \(\rho : \Mod(S_g^b) \to + Let \(g \ge 1\), \(p \ge 0\) and fix \(\rho : \Mod(S_g^p) \to \GL_n(\mathbb{C})\) with \(n < 2g\). As promised, we proceed by induction on \(g\). The base case \(g = 1\) is again clear from the fact \(n = 1\) and \(\GL_1(\mathbb{C}) = \mathbb{C}^\times\). The case \(g = 2\) was also established in Proposition~\ref{thm:low-dim-reps-are-trivial-base-case}. Now suppose \(g \ge 3\) and every \(m\)-dimensional representation of \(S_{g - - 1}^{b'}\) has Abelian image for \(m < 2(g - 1)\). Let us show \(\rho\) has + 1}^q\) has Abelian image for \(m < 2(g - 1)\). Let us show \(\rho\) has Abelian image. Let \(\alpha_1, \ldots, \alpha_g, \beta_1, \ldots, \beta_g, \gamma_1, \ldots, - \gamma_{g - 1}, \eta_1, \ldots, \eta_{b - 1} \subset S_g^b\) be the curves - from the Lickorish generators of \(\Mod(S_g^b)\), as in + \gamma_{g - 1}, \eta_1, \ldots, \eta_{p-1} \subset S_g^p\) be the curves + from the Lickorish generators of \(\Mod(S_g^p)\), as in Figure~\ref{fig:lickorish-gens}. Once again, let \(L_\alpha = \rho(\tau_\alpha)\) and denote by \(E_{\alpha = \lambda}\) the eigenspace of \(L_\alpha\) associated to \(\lambda \in \mathbb{C}\). Let \(R \cong S_{g - @@ -256,14 +256,14 @@ representations. \begin{figure}[ht] \centering \includegraphics[width=.35\linewidth]{images/lickorish-gens-korkmaz-proof.eps} - \caption{The subsurface $R \subset S_g^b$.} + \caption{The subsurface $R \subset S_g^p$.} \label{fig:korkmaz-proof-subsurface} \end{figure} % TODO: Add more comments on the injectivity of this map? We claim that it suffices to find a \(m\)-dimensional \(\Mod(R)\)-invariant\footnote{Here we view $\Mod(R)$ as a subgroup of - $\Mod(S_g^b)$ via the inclusion homomorphism $\Mod(R) \to \Mod(S_g^b)$ from + $\Mod(S_g^p)$ via the inclusion homomorphism $\Mod(R) \to \Mod(S_g^p)$ from Example~\ref{ex:inclusion-morphism}, which can be shown to be injective in this particular case.} subspace \(W \subset \mathbb{C}^n\) with \(2 \le m \le n - 2\). Indeed, in this case \(m < 2(g - 1)\) and \(\dim @@ -291,9 +291,9 @@ representations. follows from Proposition~\ref{thm:commutator-is-perfect} that \(\rho\) annihilates all of \(\Mod(R)'\) and, in particular, \(\tau_{\alpha_1} \tau_{\beta_1}^{-1} \in \ker \rho\). But recall from - Proposition~\ref{thm:commutator-normal-gen} that \(\Mod(S_g^b)'\) is normally + Proposition~\ref{thm:commutator-normal-gen} that \(\Mod(S_g^p)'\) is normally generated by \(\tau_{\alpha_1} \tau_{\beta_1}^{-1}\), from which we conclude - \(\rho(\Mod(S_g^b)') = 1\), as desired. + \(\rho(\Mod(S_g^p)') = 1\), as desired. As before, we exhaustively analyse all possible Jordan forms of \(L_{\alpha_g}\). First, consider the case where we can find eigenvalues @@ -354,9 +354,9 @@ representations. For case (1), we use the change of coordinates principle: each \(L_{\alpha_i}, L_{\beta_i}, L_{\gamma_i}, L_{\eta_i}\) is conjugate to - \(L_{\alpha_g} = \lambda\), so all Lickorish generators of \(\Mod(S_g^b)\) + \(L_{\alpha_g} = \lambda\), so all Lickorish generators of \(\Mod(S_g^p)\) act on \(\mathbb{C}^n\) as scalar multiplication by \(\lambda\) as well. - Hence \(\rho(\Mod(S_g^b))\) is cyclic and thus Abelian. In case (2), \(W = + Hence \(\rho(\Mod(S_g^p))\) is cyclic and thus Abelian. In case (2), \(W = \ker (L_{\alpha_g} - \lambda)^2\) is a \(2\)-dimensional \(\Mod(R)\)-invariant subspace. @@ -372,7 +372,7 @@ representations. E_{\beta_g = \lambda}\). In this situation, as in the proof of Proposition~\ref{thm:low-dim-reps-are-trivial-base-case} it follows from the change of coordinates principle that there are \(f_i, g_i, h_i \in - \Mod(S_g^b)\) with + \Mod(S_g^p)\) with \begin{align*} f_i \tau_{\alpha_g} f_i^{-1} & = \tau_{\alpha_i} & @@ -391,7 +391,7 @@ representations. E_{\alpha_1 = \lambda} = \cdots = E_{\alpha_g = \lambda} = E_{\beta_1 = \lambda} = \cdots = E_{\beta_g = \lambda} = E_{\gamma_1 = \lambda} = \cdots = E_{\gamma_{g - 1} = \lambda} - = E_{\eta_1 = \lambda} = \cdots = E_{\eta_{b - 1} = \lambda}. + = E_{\eta_1 = \lambda} = \cdots = E_{\eta_{p-1} = \lambda}. \] In particular, we can find a basis for \(\mathbb{C}^n\) with respect to which @@ -405,21 +405,21 @@ representations. 0 & 0 & \cdots & 0 & * \end{pmatrix}. \] - Since the group of upper triangular matrices is solvable and \(\Mod(S_g^b)\) - is perfect, it follows that \(\rho(\Mod(S_g^b))\) is trivial. This concludes - the proof \(\rho(\Mod(S_g^b))\) is Abelian. - - To see that \(\rho(\Mod(S_g^b)) = 1\) for \(g \ge 3\) we note that, since - \(\rho(\Mod(S_g^b))\) is Abelian, \(\rho\) factors though the Abelinization - map \(\Mod(S_g^b) \to \Mod(S_g^b)^\ab = \mfrac{\Mod(S_g^b)}{[\Mod(S_g^b), - \Mod(S_g^b)]}\). Now recall from Proposition~\ref{thm:trivial-abelianization} - that \(\Mod(S_g^b)^\ab = 0\) for \(g \ge 3\). In other words, \(\rho\) + Since the group of upper triangular matrices is solvable and \(\Mod(S_g^p)\) + is perfect, it follows that \(\rho(\Mod(S_g^p))\) is trivial. This concludes + the proof \(\rho(\Mod(S_g^p))\) is Abelian. + + To see that \(\rho(\Mod(S_g^p)) = 1\) for \(g \ge 3\) we note that, since + \(\rho(\Mod(S_g^p))\) is Abelian, \(\rho\) factors though the Abelinization + map \(\Mod(S_g^p) \to \Mod(S_g^p)^\ab = \mfrac{\Mod(S_g^p)}{[\Mod(S_g^p), + \Mod(S_g^p)]}\). Now recall from Proposition~\ref{thm:trivial-abelianization} + that \(\Mod(S_g^p)^\ab = 0\) for \(g \ge 3\). In other words, \(\rho\) factors though the homomorphism \(1 \to \GL_n(\mathbb{C})\). We are done. \end{proof} Having established the triviality of the low-dimensional representations \(\rho -: \Mod(S_g^b) \to \GL_n(\mathbb{C})\), all that remains for us is to understand -the \(2g\)-dimensional reprensentations of \(\Mod(S_g^b)\). We certainly know a +: \Mod(S_g^p) \to \GL_n(\mathbb{C})\), all that remains for us is to understand +the \(2g\)-dimensional reprensentations of \(\Mod(S_g^p)\). We certainly know a nontrivial example of such, namely the symplectic representation \(\psi : \Mod(S_g) \to \operatorname{Sp}_{2g}(\mathbb{Z})\) from Example~\ref{ex:symplectic-rep}. Surprinsgly, this turns out to be @@ -427,13 +427,13 @@ Example~\ref{ex:symplectic-rep}. Surprinsgly, this turns out to be representation in the compact case. More precisely, \begin{theorem}[Korkmaz]\label{thm:reps-of-dim-2g-are-symplectic} - Let \(g \ge 3\) and \(\rho : \Mod(S_g^b) \to \GL_{2g}(\mathbb{C})\). Then + Let \(g \ge 3\) and \(\rho : \Mod(S_g^p) \to \GL_{2g}(\mathbb{C})\). Then \(\rho\) is either trivial or conjugate to the symplectic - representation\footnote{Here the map $\Mod(S_g^b) \to + representation\footnote{Here the map $\Mod(S_g^p) \to \operatorname{Sp}_{2g}(\mathbb{Z})$ is given by the composition of the - inclusion morphism $\Mod(S_g^b) \to \Mod(S_g)$ with the usual symplect + inclusion morphism $\Mod(S_g^p) \to \Mod(S_g)$ with the usual symplect representation $\psi : \Mod(S_g) \to \operatorname{Sp}_{2g}(\mathbb{Z})$.} - \(\Mod(S_g^b) \to \operatorname{Sp}_{2g}(\mathbb{Z})\) of \(\Mod(S_g^b)\). + \(\Mod(S_g^p) \to \operatorname{Sp}_{2g}(\mathbb{Z})\) of \(\Mod(S_g^p)\). \end{theorem} Unfortunately, the limited scope of these master thesis does not allow us to @@ -483,7 +483,7 @@ to as \emph{the main lemma}. Namely\dots This is proved in \cite[Lemma 7.6]{korkmaz} using the braid relations. Notice that for \(n = g\) and \(m = 2g\) the matrices in Lemma~\ref{thm:main-lemma} coincide with the action of the Lickrish generators \(\tau_{\alpha_1}, \ldots, -\tau_{\alpha_g}, \tau_{\beta_1}, \ldots, \tau_{\beta_g} \in \Mod(S_g^b)\) on +\tau_{\alpha_g}, \tau_{\beta_1}, \ldots, \tau_{\beta_g} \in \Mod(S_g^p)\) on \(H_1(S_g, \mathbb{C}) \cong \mathbb{C}^{2g}\) -- represented in the standard basis \([\alpha_1], \ldots, [\alpha_g], [\beta_1], \ldots, [\beta_g]\) for \(H_1(S_g, \mathbb{C})\). @@ -506,28 +506,28 @@ basis \([\alpha_1], \ldots, [\alpha_g], [\beta_1], \ldots, [\beta_g]\) for \right) \end{align*} -Hence by embeding \(B_3^g\) in \(\Mod(S_g^b)\) via +Hence by embeding \(B_3^g\) in \(\Mod(S_g^p)\) via \begin{align*} - B_3^g & \to \Mod(S_g^b) \\ + B_3^g & \to \Mod(S_g^p) \\ a_i & \mapsto \tau_{\alpha_i} \\ b_i & \mapsto \tau_{\beta_i} \end{align*} -we can see that any \(\rho : \Mod(S_g^b) \to \GL_{2g}(\mathbb{C})\) in a +we can see that any \(\rho : \Mod(S_g^p) \to \GL_{2g}(\mathbb{C})\) in a certain class of representation satisfying some technical conditions must be -conjugate to the symplectic representation \(\Mod(S_g^b) \to +conjugate to the symplectic representation \(\Mod(S_g^p) \to \operatorname{Sp}_{2g}(\mathbb{Z})\) when restricted to \(B_3^g\). Korkmaz then goes on to show that such technical conditions are met for any -nontrivial \(\rho : \Mod(S_g^b) \to \GL_{2g}(\mathbb{C})\). Furthermore, +nontrivial \(\rho : \Mod(S_g^p) \to \GL_{2g}(\mathbb{C})\). Furthermore, Korkmaz also argues that we can find a basis for \(\mathbb{C}^{2g}\) with respect to which the matrices of \(\rho(\tau_{\gamma_1}), \ldots, \rho(\tau_{\gamma_{g - 1}}), \rho(\tau_{\eta_1}), \ldots, -\rho(\tau_{\eta_{b-1}})\) also agrees with the action of \(\Mod(S_g^b)\) on +\rho(\tau_{\eta_{p-1}})\) also agrees with the action of \(\Mod(S_g^p)\) on \(H_1(S_g, \mathbb{C})\), concluding the classification of \(2g\)-dimensional representations. % TODO: Add some final comments about how the rest of the landscape of % representations is generally unknown and how there is a lot to study in here Recently, Kasahara \cite{kasahara} also classified the \((2g+1)\)-dimensional -representations of \(\Mod(S_g^b)\) for \(g \ge 7\) in terms of certain twisted +representations of \(\Mod(S_g^p)\) for \(g \ge 7\) in terms of certain twisted \(1\)-cohomology groups.
diff --git a/sections/twists.tex b/sections/twists.tex @@ -183,9 +183,9 @@ too. A perhaps less obvious fact about Dehn twists is\dots \begin{theorem}\label{thm:mcg-is-fg} - Let \(S_{g, r}^b\) be the orientable surface of genus \(g \ge 1\) with \(r\) + Let \(S_{g, r}^p\) be the orientable surface of genus \(g \ge 1\) with \(r\) punctures and \(b\) boundary components. Then the pure mapping class group - \(\PMod(S_{g, r}^b)\) is generated by finitely many Dehn twists about + \(\PMod(S_{g, r}^p)\) is generated by finitely many Dehn twists about nonseparating curves or boundary components. \end{theorem} @@ -198,9 +198,9 @@ curves}. \section{The Birman Exact Sequence} Having the proof of Theorem~\ref{thm:mcg-is-fg} in mind, it is interesting to -consider the relationship between the mapping class group of \(S_{g, r}^b\) and -that of \(S_{g, r+1}^b = S_{g, r}^b \setminus \{ x \}\) for some \(x\) in the -interior \((S_{g, r}^b)\degree\) of \(S_{g, r}^b\). Indeed, this will later +consider the relationship between the mapping class group of \(S_{g, r}^p\) and +that of \(S_{g, r+1}^p = S_{g, r}^p \setminus \{ x \}\) for some \(x\) in the +interior \((S_{g, r}^p)\degree\) of \(S_{g, r}^p\). Indeed, this will later allow us to establish the induction on the number of punctures \(r\). Given an orientable surface \(S\) and \(x_1, \ldots, x_n \in S\degree\), @@ -289,7 +289,7 @@ show\dots \section{The Modified Graph of Curves} Having established Theorem~\ref{thm:birman-exact-seq}, we now need to adress -the induction step in the genus \(g\) of \(S_{g, r}^b\). Our strategy is to +the induction step in the genus \(g\) of \(S_{g, r}^p\). Our strategy is to apply the following lemma from geomtric group theory. \begin{lemma}\label{thm:ggt-lemma} @@ -301,7 +301,7 @@ apply the following lemma from geomtric group theory. \(G\) is generated by \(G_v\) and \(g\). \end{lemma} -We are interested, of course, in the group \(G = \PMod(S_{g, r}^b)\). As for +We are interested, of course, in the group \(G = \PMod(S_{g, r}^p)\). As for the graph \(\Gamma\), we consider\dots \begin{definition} @@ -317,9 +317,9 @@ the graph \(\Gamma\), we consider\dots \end{definition} It is clear from Lemma~\ref{thm:change-of-coordinates} that the actions of -\(\Mod(S_{g, r}^b)\) on \(V(\hat{\mathcal{N}}(S_{g, r}^b))\) and \(\{([\alpha], -[\beta]) \in V(\hat{\mathcal{N}}(S_{g, r}^b))^2 : \#(\alpha \cap \beta) = 1 -\}\) are both transitive. But why should \(\hat{\mathcal{N}}(S_{g, r}^b)\) be +\(\Mod(S_{g, r}^p)\) on \(V(\hat{\mathcal{N}}(S_{g, r}^p))\) and \(\{([\alpha], +[\beta]) \in V(\hat{\mathcal{N}}(S_{g, r}^p))^2 : \#(\alpha \cap \beta) = 1 +\}\) are both transitive. But why should \(\hat{\mathcal{N}}(S_{g, r}^p)\) be connected? Historically, the modified graph of nonseparating curves first arised as a @@ -361,30 +361,30 @@ Corollary~\ref{thm:mofied-graph-is-connected}. We are now ready to show Theorem~\ref{thm:mcg-is-fg}. \begin{proof}[Proof of Theorem~\ref{thm:mcg-is-fg}] - Let \(S_{g, r}^b\) be the orientable surface of genus \(g \ge 1\) with \(r\) + Let \(S_{g, r}^p\) be the orientable surface of genus \(g \ge 1\) with \(r\) punctures and \(b\) boundary components. We want to establish that - \(\PMod(S_{g, r}^b)\) is genetery by a finite number of Dehn twists about + \(\PMod(S_{g, r}^p)\) is genetery by a finite number of Dehn twists about nonseparating simple closed curves or boundary components. - First, observe that if \(b \ge 1\) and \(\partial S_{g, r}^b = \delta_1 \cup - \cdots \cup \delta_b\) then, by recursively applying the capping exact + First, observe that if \(p \ge 1\) and \(\partial S_{g, r}^p = \delta_1 \cup + \cdots \cup \delta_p\) then, by recursively applying the capping exact sequence \begin{center} \begin{tikzcd} 1 \rar & \langle \tau_{\delta_1} \rangle \rar & - \PMod(S_{g, r}^b) \rar{\operatorname{cap}} & - \PMod(S_{g, r}^b \cup_{\delta_1} (\mathbb{D}^2 \setminus \{0\})) \rar & + \PMod(S_{g, r}^p) \rar{\operatorname{cap}} & + \PMod(S_{g, r}^p \cup_{\delta_1} (\mathbb{D}^2 \setminus \{0\})) \rar & 1 \end{tikzcd} \end{center} from Example~\ref{ex:capping-seq}, it suffices to show that \(S_{g, n}\) is finitely generated by twists about nonseparating simple closed curves. - Indeed, if \(\PMod(S_{g, r}^b \cup_{\delta_1} (\mathbb{D}^2 \setminus \{0\}))\) + Indeed, if \(\PMod(S_{g, r}^p \cup_{\delta_1} (\mathbb{D}^2 \setminus \{0\}))\) is finitely generated by twists about nonseparing curves or boundary - components, then we may lift the generators of \(\PMod(S_{g, r}^b + components, then we may lift the generators of \(\PMod(S_{g, r}^p \cup_{\delta_1} (\mathbb{D}^2 \setminus \{0\}))\) to Dehn twists about the - corresponding curves in \(S_{g, r}^b\) and add \(\tau_{\delta_1}\) to the + corresponding curves in \(S_{g, r}^p\) and add \(\tau_{\delta_1}\) to the generating set. It thus suffices to consider the boudaryless case \(S_{g, r}\). As promised, @@ -498,18 +498,18 @@ Theorem~\ref{thm:mcg-is-fg}. There are many possible improvements to this last result. For instance, in \cite[Section~4.4]{farb-margalit} Farb-Margalit exhibit an explicit set of -generators of \(\Mod(S_g^b)\) by addapting the induction steps in the proof of +generators of \(\Mod(S_g^p)\) by addapting the induction steps in the proof of Theorem~\ref{thm:mcg-is-fg}. These are known as the \emph{Lickorish generators}. \begin{theorem}[Lickorish generators]\label{thm:lickorish-gens} - If \(g \ge 1\) then \(\Mod(S_g^b)\) is generated by the Dehn twists about the + If \(g \ge 1\) then \(\Mod(S_g^p)\) is generated by the Dehn twists about the curves \(\alpha_1, \ldots, \alpha_g, \beta_1, \ldots, \beta_g, \gamma_1, - \ldots, \gamma_{g - 1}, \eta_1, \ldots, \eta_{b - 1}\) as in + \ldots, \gamma_{g - 1}, \eta_1, \ldots, \eta_{p-1}\) as in Figure~\ref{fig:lickorish-gens} \end{theorem} -In the boundaryless case \(b = 0\), we can write \(\tau_{\mu_3}, \ldots, +In the boundaryless case \(p = 0\), we can write \(\tau_{\mu_3}, \ldots, \tau_{\mu_g} \in \Mod(S_g)\) as products of the twists about the remaining curves, from which we get the so called \emph{Humphreys generators}. @@ -522,7 +522,7 @@ curves, from which we get the so called \emph{Humphreys generators}. \begin{minipage}[b]{.45\linewidth} \centering \includegraphics[width=\linewidth]{images/lickorish-gens.eps} - \captionof{figure}{The curves from Lickorish generators of $\Mod(S_g^b)$.} + \captionof{figure}{The curves from Lickorish generators of $\Mod(S_g^p)$.} \label{fig:lickorish-gens} \end{minipage} \hspace{.5cm} %