553:
468:
900:
265:
1130:
1057:
347:
824:
555:
and endow this set with the topology of pointwise convergence (sometimes called "algebraic convergence"). In this particular case this topology can most easily be defined as follows: the group
742:
1259:
201:
443:
293:
950:
678:
643:
93:
which is not isomorphic to either the
Riemann sphere (the elliptic case) or a quotient of the complex plane by a discrete subgroup (the parabolic case) must be a quotient of the
1083:
1323:
1000:
116:
1213:
771:
573:
415:
136:
1291:
1177:
1150:
978:
597:
463:
395:
367:
313:
91:
66:
829:
548:{\displaystyle A(\Gamma )=\{\rho \colon \Gamma \to \mathrm {PSL} _{2}(\mathbb {R} )\colon \rho {\text{ is faithful and discrete }}\}}
218:
1092:
1005:
318:
776:
139:
1367:
154:
150:
683:
1218:
160:
420:
270:
1372:
910:
576:
58:
920:
648:
602:
1062:
1156:
1296:
1344:
70:
983:
99:
1182:
957:
212:
747:
1132:
and lift it to the hyperbolic plane. Taking a diffeomorphism yields quasi-conformal map since
62:
558:
400:
121:
94:
45:. Every hyperbolic Riemann surface admits such a representation. The concept is named after
35:
1264:
208:
28:
1355:
Matsuzaki, K.; Taniguchi, M.: Hyperbolic manifolds and
Kleinian groups. Oxford (1998).
1335:
1162:
1135:
963:
582:
448:
380:
352:
298:
76:
42:
1361:
953:
46:
909:(this is not standard terminology and this result is not directly related to the
1339:
143:
20:
204:
1325:
is a quasiconformal homeomorphism modulo a natural equivalence relation.
1179:: the set of discrete faithful representations of the fundamental group
1155:
This result can be seen as the equivalence between two models for
895:{\displaystyle \rho \mapsto (\rho (g_{1}),\ldots ,\rho (g_{r}))}
315:. Such a group is called a Fuchsian group, and the isomorphism
260:{\displaystyle \Gamma \subset \mathrm {PSL} _{2}(\mathbb {R} )}
1114:
427:
331:
277:
73:. More precisely this theorem states that a Riemann surface
207:, and the uniformization theorem means that there exists a
1125:{\displaystyle R\to \rho (\Gamma )\backslash \mathbb {H} }
1052:{\displaystyle h\circ \gamma \circ h^{-1}=\rho (\gamma )}
1261:
modulo conjugacy and the set of marked
Riemann surfaces
1299:
1267:
1221:
1185:
1165:
1138:
1095:
1065:
1008:
986:
966:
923:
832:
779:
750:
686:
651:
605:
585:
561:
471:
451:
423:
403:
383:
355:
342:{\displaystyle R\cong \Gamma \backslash \mathbb {H} }
321:
301:
273:
221:
163:
124:
102:
79:
819:{\displaystyle \mathrm {PSL} _{2}(\mathbb {R} )^{r}}
579:since it is isomorphic to the fundamental group of
1317:
1285:
1253:
1207:
1171:
1144:
1124:
1089:The proof is very simple: choose an homeomorphism
1077:
1051:
994:
972:
944:
894:
818:
765:
736:
672:
637:
591:
567:
547:
457:
437:
409:
389:
361:
341:
307:
287:
259:
195:
130:
110:
85:
737:{\displaystyle \rho (g_{1}),\ldots ,\rho (g_{r})}
1254:{\displaystyle \mathrm {PSL} _{2}(\mathbb {R} )}
196:{\displaystyle \mathrm {PSL} _{2}(\mathbb {R} )}
438:{\displaystyle \Gamma \backslash \mathbb {H} }
288:{\displaystyle \Gamma \backslash \mathbb {H} }
8:
542:
487:
902:. Then we give it the subspace topology.
1298:
1266:
1244:
1243:
1234:
1223:
1220:
1190:
1184:
1164:
1137:
1118:
1117:
1094:
1064:
1025:
1007:
988:
987:
985:
965:
922:
880:
852:
831:
810:
802:
801:
792:
781:
778:
749:
725:
697:
685:
650:
629:
610:
604:
584:
560:
537:
524:
523:
514:
503:
470:
450:
431:
430:
422:
402:
382:
354:
335:
334:
320:
300:
281:
280:
272:
250:
249:
240:
229:
220:
186:
185:
176:
165:
162:
123:
104:
103:
101:
78:
16:Group representation of a Riemann surface
397:be a closed hyperbolic surface and let
153:for the hyperbolic plane the group of
913:) then has the following statement:
373:Fuchsian models and TeichmĂĽller space
7:
539: is faithful and discrete
27:is a representation of a hyperbolic
945:{\displaystyle \rho \in A(\Gamma )}
673:{\displaystyle \rho \in A(\Gamma )}
638:{\displaystyle g_{1},\ldots ,g_{r}}
1230:
1227:
1224:
1108:
1078:{\displaystyle \gamma \in \Gamma }
1072:
936:
788:
785:
782:
757:
664:
562:
510:
507:
504:
496:
478:
424:
404:
328:
274:
236:
233:
230:
222:
172:
169:
166:
125:
61:, every Riemann surface is either
14:
1338:, an analogous construction for
349:is called a Fuchsian model for
1318:{\displaystyle f\colon R\to X}
1309:
1280:
1268:
1248:
1240:
1202:
1196:
1111:
1105:
1099:
1046:
1040:
939:
933:
889:
886:
873:
858:
845:
839:
836:
807:
798:
760:
754:
731:
718:
703:
690:
680:is determined by the elements
667:
661:
645:be a generating set: then any
528:
520:
499:
481:
475:
267:such that the Riemann surface
254:
246:
190:
182:
1:
155:biholomorphic transformations
995:{\displaystyle \mathbb {H} }
417:be a Fuchsian group so that
111:{\displaystyle \mathbb {H} }
1208:{\displaystyle \pi _{1}(R)}
907:Nielsen isomorphism theorem
1389:
766:{\displaystyle A(\Gamma )}
151:Poincaré half-plane model
53:A more precise definition
980:of the upper half-plane
445:is a Fuchsian model for
140:properly discontinuously
744:and so we can identify
568:{\displaystyle \Gamma }
410:{\displaystyle \Gamma }
131:{\displaystyle \Gamma }
1319:
1287:
1255:
1209:
1173:
1146:
1126:
1079:
1053:
996:
974:
946:
896:
820:
767:
738:
674:
639:
593:
569:
549:
459:
439:
411:
391:
363:
343:
309:
289:
261:
197:
132:
112:
87:
59:uniformization theorem
1320:
1288:
1286:{\displaystyle (X,f)}
1256:
1210:
1174:
1147:
1127:
1080:
1054:
997:
975:
947:
897:
821:
768:
739:
675:
640:
594:
570:
550:
460:
440:
412:
392:
364:
344:
310:
290:
262:
198:
133:
113:
88:
34:as a quotient of the
1297:
1265:
1219:
1183:
1163:
1136:
1093:
1063:
1006:
984:
964:
952:there exists a self-
921:
911:Dehn–Nielsen theorem
830:
777:
748:
684:
649:
603:
583:
559:
469:
449:
421:
401:
381:
353:
319:
299:
271:
219:
161:
122:
100:
77:
1368:Hyperbolic geometry
1345:Fundamental polygon
1315:
1283:
1251:
1205:
1169:
1142:
1122:
1075:
1049:
992:
970:
958:quasiconformal map
942:
892:
816:
763:
734:
670:
635:
589:
577:finitely generated
565:
545:
455:
435:
407:
387:
359:
339:
305:
285:
257:
193:
128:
108:
83:
1172:{\displaystyle R}
1157:TeichmĂĽller space
1145:{\displaystyle R}
973:{\displaystyle h}
773:with a subset of
592:{\displaystyle R}
540:
458:{\displaystyle R}
390:{\displaystyle R}
362:{\displaystyle R}
308:{\displaystyle R}
295:is isomorphic to
86:{\displaystyle R}
1380:
1373:Riemann surfaces
1324:
1322:
1321:
1316:
1292:
1290:
1289:
1284:
1260:
1258:
1257:
1252:
1247:
1239:
1238:
1233:
1214:
1212:
1211:
1206:
1195:
1194:
1178:
1176:
1175:
1170:
1151:
1149:
1148:
1143:
1131:
1129:
1128:
1123:
1121:
1084:
1082:
1081:
1076:
1058:
1056:
1055:
1050:
1033:
1032:
1001:
999:
998:
993:
991:
979:
977:
976:
971:
951:
949:
948:
943:
901:
899:
898:
893:
885:
884:
857:
856:
825:
823:
822:
817:
815:
814:
805:
797:
796:
791:
772:
770:
769:
764:
743:
741:
740:
735:
730:
729:
702:
701:
679:
677:
676:
671:
644:
642:
641:
636:
634:
633:
615:
614:
598:
596:
595:
590:
574:
572:
571:
566:
554:
552:
551:
546:
541:
538:
527:
519:
518:
513:
464:
462:
461:
456:
444:
442:
441:
436:
434:
416:
414:
413:
408:
396:
394:
393:
388:
368:
366:
365:
360:
348:
346:
345:
340:
338:
314:
312:
311:
306:
294:
292:
291:
286:
284:
266:
264:
263:
258:
253:
245:
244:
239:
202:
200:
199:
194:
189:
181:
180:
175:
137:
135:
134:
129:
117:
115:
114:
109:
107:
95:hyperbolic plane
92:
90:
89:
84:
36:upper half-plane
1388:
1387:
1383:
1382:
1381:
1379:
1378:
1377:
1358:
1357:
1353:
1331:
1295:
1294:
1263:
1262:
1222:
1217:
1216:
1186:
1181:
1180:
1161:
1160:
1134:
1133:
1091:
1090:
1087:
1061:
1060:
1021:
1004:
1003:
982:
981:
962:
961:
919:
918:
876:
848:
828:
827:
806:
780:
775:
774:
746:
745:
721:
693:
682:
681:
647:
646:
625:
606:
601:
600:
581:
580:
557:
556:
502:
467:
466:
447:
446:
419:
418:
399:
398:
379:
378:
375:
351:
350:
317:
316:
297:
296:
269:
268:
228:
217:
216:
164:
159:
158:
120:
119:
98:
97:
75:
74:
55:
29:Riemann surface
17:
12:
11:
5:
1386:
1384:
1376:
1375:
1370:
1360:
1359:
1352:
1349:
1348:
1347:
1342:
1336:Kleinian model
1330:
1327:
1314:
1311:
1308:
1305:
1302:
1282:
1279:
1276:
1273:
1270:
1250:
1246:
1242:
1237:
1232:
1229:
1226:
1204:
1201:
1198:
1193:
1189:
1168:
1141:
1120:
1116:
1113:
1110:
1107:
1104:
1101:
1098:
1074:
1071:
1068:
1048:
1045:
1042:
1039:
1036:
1031:
1028:
1024:
1020:
1017:
1014:
1011:
990:
969:
941:
938:
935:
932:
929:
926:
915:
891:
888:
883:
879:
875:
872:
869:
866:
863:
860:
855:
851:
847:
844:
841:
838:
835:
813:
809:
804:
800:
795:
790:
787:
784:
762:
759:
756:
753:
733:
728:
724:
720:
717:
714:
711:
708:
705:
700:
696:
692:
689:
669:
666:
663:
660:
657:
654:
632:
628:
624:
621:
618:
613:
609:
588:
564:
544:
536:
533:
530:
526:
522:
517:
512:
509:
506:
501:
498:
495:
492:
489:
486:
483:
480:
477:
474:
454:
433:
429:
426:
406:
386:
374:
371:
358:
337:
333:
330:
327:
324:
304:
283:
279:
276:
256:
252:
248:
243:
238:
235:
232:
227:
224:
192:
188:
184:
179:
174:
171:
168:
127:
118:by a subgroup
106:
82:
54:
51:
43:Fuchsian group
25:Fuchsian model
15:
13:
10:
9:
6:
4:
3:
2:
1385:
1374:
1371:
1369:
1366:
1365:
1363:
1356:
1350:
1346:
1343:
1341:
1337:
1333:
1332:
1328:
1326:
1312:
1306:
1303:
1300:
1277:
1274:
1271:
1235:
1199:
1191:
1187:
1166:
1158:
1153:
1152:is compact.
1139:
1102:
1096:
1086:
1069:
1066:
1043:
1037:
1034:
1029:
1026:
1022:
1018:
1015:
1012:
1009:
967:
959:
955:
954:homeomorphism
930:
927:
924:
914:
912:
908:
903:
881:
877:
870:
867:
864:
861:
853:
849:
842:
833:
811:
793:
751:
726:
722:
715:
712:
709:
706:
698:
694:
687:
658:
655:
652:
630:
626:
622:
619:
616:
611:
607:
586:
578:
534:
531:
515:
493:
490:
484:
472:
452:
384:
372:
370:
356:
325:
322:
302:
241:
225:
214:
210:
206:
177:
157:is the group
156:
152:
147:
145:
141:
96:
80:
72:
68:
64:
60:
52:
50:
48:
47:Lazarus Fuchs
44:
40:
37:
33:
30:
26:
22:
1354:
1154:
1088:
916:
906:
904:
376:
213:torsion-free
205:homographies
148:
56:
38:
31:
24:
18:
1340:3-manifolds
956:(in fact a
826:by the map
21:mathematics
1362:Categories
1351:References
1002:such that
203:acting by
71:hyperbolic
1310:→
1304::
1188:π
1115:∖
1109:Γ
1103:ρ
1100:→
1073:Γ
1070:∈
1067:γ
1044:γ
1038:ρ
1027:−
1019:∘
1016:γ
1013:∘
937:Γ
928:∈
925:ρ
871:ρ
865:…
843:ρ
837:↦
834:ρ
758:Γ
716:ρ
710:…
688:ρ
665:Γ
656:∈
653:ρ
620:…
563:Γ
535:ρ
532::
500:→
497:Γ
494::
491:ρ
479:Γ
428:∖
425:Γ
405:Γ
332:∖
329:Γ
326:≅
278:∖
275:Γ
226:⊂
223:Γ
215:subgroup
126:Γ
67:parabolic
1329:See also
1059:for all
917:For any
209:discrete
63:elliptic
465:. Let
149:In the
138:acting
57:By the
1293:where
599:. Let
144:freely
1215:into
41:by a
1334:the
905:The
377:Let
142:and
23:, a
1159:of
575:is
146:.
69:or
19:In
1364::
960:)
369:.
211:,
65:,
49:.
1313:X
1307:R
1301:f
1281:)
1278:f
1275:,
1272:X
1269:(
1249:)
1245:R
1241:(
1236:2
1231:L
1228:S
1225:P
1203:)
1200:R
1197:(
1192:1
1167:R
1140:R
1119:H
1112:)
1106:(
1097:R
1085:.
1047:)
1041:(
1035:=
1030:1
1023:h
1010:h
989:H
968:h
940:)
934:(
931:A
890:)
887:)
882:r
878:g
874:(
868:,
862:,
859:)
854:1
850:g
846:(
840:(
812:r
808:)
803:R
799:(
794:2
789:L
786:S
783:P
761:)
755:(
752:A
732:)
727:r
723:g
719:(
713:,
707:,
704:)
699:1
695:g
691:(
668:)
662:(
659:A
631:r
627:g
623:,
617:,
612:1
608:g
587:R
543:}
529:)
525:R
521:(
516:2
511:L
508:S
505:P
488:{
485:=
482:)
476:(
473:A
453:R
432:H
385:R
357:R
336:H
323:R
303:R
282:H
255:)
251:R
247:(
242:2
237:L
234:S
231:P
191:)
187:R
183:(
178:2
173:L
170:S
167:P
105:H
81:R
39:H
32:R
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