1189:
The type of a
Fuchsian group need not be the same as its type when considered as a Kleinian group: in fact, all Fuchsian groups are Kleinian groups of type 2, as their limit sets (as Kleinian groups) are proper subsets of the Riemann sphere, contained in some circle.
454:
1305:
209:
870:
Although discontinuity and discreteness are equivalent in this case, this is not generally true for the case of an arbitrary group of conformal homeomorphisms acting on the full
Riemann sphere (as opposed to
1415:
1600:
312:
636:
1065:
That is, any one of these three can serve as a definition of a
Fuchsian group, the others following as theorems. The notion of an invariant proper subset Δ is important; the so-called
925:
1646:"Ueber eine Klasse von Funktionen mehrerer Variablen, welche durch Umkehrung der Integrale von Lösungen der linearen Differentialgleichungen mit rationalen Coeffizienten entstehen"
811:
844:
783:
63:, so a Fuchsian group can be regarded as a group acting on any of these spaces. There are some variations of the definition: sometimes the Fuchsian group is assumed to be
741:
1182:∪ ∞. Since it is nowhere dense, this implies that any limit point is arbitrarily close to an open set that is not in the limit set. In other words, the limit set is a
761:
704:
684:
511:
1058:
955:
362:
1216:
889:
864:
660:
583:
563:
539:
488:
346:
233:
1144:∪ ∞. The limit set may be empty, or may contain one or two points, or may contain an infinite number. In the latter case, there are two types:
152:
1089:. Similarly, the idea that Δ is a proper subset of the region of discontinuity is important; when it is not, the subgroup is called a
1367:
763:
converging to the identity in the usual topology of point-wise convergence is eventually constant, i.e. there exists an integer
1541:
245:
591:
1764:
1746:
1723:
1703:
1677:
1623:
1463:
1467:
1478:
349:
1694:
1827:
1669:
898:
1848:
1085:
a
Fuchsian group, does not act discontinuously on the real number line; it has accumulation points at the
1822:
788:
586:
325:
1853:
1817:
1759:, (1989) London Mathematical Society Lecture Note Series 143, Cambridge University Press, Cambridge
1858:
1843:
1737:
1047:
816:
64:
56:
1358:
766:
353:
1000:
to some open disk Δ. Conjugating by such a transformation will send a discrete subgroup of PSL(2,
719:
1645:
1528:
1428:
1124:
in the upper half-plane. There may, however, be limit points on the real axis. Let Λ(Γ) be the
107:
103:
746:
689:
669:
496:
1609:
of the corresponding
Riemann surface, if the Fuchsian group is torsion-free and co-compact.
1159:/Γ has finite volume, but there are Fuchsian groups of the first kind of infinite covolume.
71:) (so that it contains orientation-reversing elements), and sometimes it is allowed to be a
1751:(Provides an excellent exposition of theory and results, richly illustrated with diagrams.)
1618:
449:{\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}\cdot z={\frac {az+b}{cz+d}}.}
1806:
1300:{\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}\cdot z={\frac {az+b}{cz+d}}}
8:
934:
52:
44:
1121:
874:
849:
645:
639:
568:
548:
524:
473:
331:
218:
111:
1073:) is discrete but does not preserve any disk in the Riemann sphere. Indeed, even the
1863:
1794:
1760:
1742:
1719:
1699:
1673:
1606:
1770:
126:
1802:
1784:
1775:
1436:
1101:
236:
212:
60:
1732:
1086:
95:
1813:
1711:
1685:
1628:
1448:
1444:
1339:
1097:
1090:
1036:
982:
708:
91:
72:
28:
1837:
1798:
1728:
1641:
1485:
1440:
1432:
1199:
1175:
1074:
138:
16:
Discrete subgroup of the real projective special linear group of dimension 2
1451:, corresponding to orientation-preserving isometries) is a Fuchsian group.
1335:
115:
1171:
20:
1789:
1503:
1183:
464:
1031:
if and only if any of the following three equivalent properties hold:
1661:
1125:
962:
1020:
467:
204:{\displaystyle H=\{z\in \mathbb {C} |\operatorname {Im} {z}>0\}}
48:
32:
977:
A linear fractional transformation defined by a matrix from PSL(2,
895:) is discrete but has accumulation points on the real number line
1096:
It is most usual to take the invariant domain Δ to be either the
318:
76:
1410:{\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}}
1170:. Equivalently, this is a group for which the limit set is a
1151:
is a group for which the limit set is the closed real line
144:
1595:{\displaystyle |\mathrm {tr} \;h|=2\cosh {\frac {L}{2}}.}
307:{\displaystyle ds={\frac {1}{y}}{\sqrt {dx^{2}+dy^{2}}}.}
1527:
of its action in the upper half-plane is related to the
1481:
cyclic subgroup is
Fuchsian if and only if it is finite.
631:{\displaystyle \Gamma z=\{\gamma z:\gamma \in \Gamma \}}
1666:
Theta
Constants, Riemann Surfaces and the Modular Group
1376:
1225:
371:
59:
of the unit disc, or conformal transformations of the
1690:
Spectral
Methods of Automorphic Forms, Second Edition
1544:
1420:
are congruent to those of the identity matrix modulo
1370:
1219:
1120:
in the upper half-plane under the action of Γ has no
937:
901:
877:
852:
819:
791:
769:
749:
722:
692:
672:
648:
594:
571:
551:
527:
499:
476:
365:
334:
248:
221:
155:
67:, sometimes it is allowed to be a subgroup of PGL(2,
1427:A co-compact example is the (ordinary, rotational)
1594:
1409:
1361:of the above form where the entries of the matrix
1299:
1210:) consisting of linear fractional transformations
949:
919:
883:
858:
838:
805:
777:
755:
735:
698:
678:
654:
630:
577:
557:
533:
505:
482:
448:
340:
306:
227:
203:
1835:
1698:), America Mathematical Society, Providence, RI
1523:is a hyperbolic element, the translation length
1502:Let Γ be a non-abelian Fuchsian group. Then the
1039:(with respect to the standard topology on PSL(2,
125:General Fuchsian groups were first studied by
1718:(1992), University of Chicago Press, Chicago
1011:This motivates the following definition of a
1345:Other Fuchsian groups include the groups Γ(
957:to every rational number, and the rationals
625:
604:
198:
162:
98:. In this case, the group may be called the
83:) which is conjugate to a subgroup of PSL(2,
1128:of Γ, that is, the set of limit points of Γ
1112:Because of the discrete action, the orbit Γ
459:This action is faithful, and in fact PSL(2,
1558:
1788:
1773:(1882), "Théorie des groupes fuchsiens",
1738:Indra's Pearls: The Vision of Felix Klein
799:
771:
513:may be defined to be a subgroup of PSL(2,
172:
1769:
1431:, containing the Fuchsian groups of the
1155:∪ ∞. This happens if the quotient space
996:∪ ∞, but will send the upper-half plane
130:
102:. In some sense, Fuchsian groups do for
1812:
145:Fuchsian groups on the upper half-plane
1836:
1198:An example of a Fuchsian group is the
920:{\displaystyle \operatorname {Im} z=0}
1757:The Ergodic Theory of Discrete Groups
1640:
972:
891:). Indeed, the Fuchsian group PSL(2,
134:
1741:, (2002) Cambridge University Press
1624:Non-Euclidean crystallographic group
1514:
463:) is isomorphic to the group of all
118:graphics are based on them (for the
43:) can be regarded equivalently as a
1491:No Fuchsian group is isomorphic to
133:), who was motivated by the paper (
90:Fuchsian groups are used to create
13:
1554:
1551:
1326:are integers. The quotient space
1206:). This is the subgroup of PSL(2,
1027:∪ ∞, that is, Γ(Δ) = Δ. Then Γ is
1004:) to a discrete subgroup of PSL(2,
750:
693:
673:
622:
595:
500:
137:), and therefore named them after
14:
1875:
1605:A similar relation holds for the
1456:Fuchsian groups of the first kind
1443:. More generally, any hyperbolic
1359:linear fractional transformations
806:{\displaystyle n>\mathbb {N} }
350:linear fractional transformations
1535:as a 2×2 matrix by the relation
1149:Fuchsian group of the first type
1695:Graduate Studies in Mathematics
1019:) act invariantly on a proper,
1783:, Springer Netherlands: 1–62,
1563:
1546:
177:
1:
1670:American Mathematical Society
1634:
1107:
1057:The set Δ is a subset of the
839:{\displaystyle \gamma _{n}=I}
666:An equivalent definition for
239:when endowed with the metric
100:Fuchsian group of the surface
778:{\displaystyle \mathbb {N} }
7:
1823:Encyclopedia of Mathematics
1612:
1447:(the index 2 subgroup of a
1193:
736:{\displaystyle \gamma _{n}}
10:
1880:
1470:cyclic subgroups of PSL(2,
47:of orientation-preserving
1488:Fuchsian group is cyclic.
122:of hyperbolic geometry).
57:conformal transformations
1048:properly discontinuously
75:(a discrete subgroup of
1692:, (2002) (Volume 53 in
1059:region of discontinuity
866:is the identity matrix.
756:{\displaystyle \Gamma }
699:{\displaystyle \Gamma }
686:to be Fuchsian is that
679:{\displaystyle \Gamma }
506:{\displaystyle \Gamma }
108:crystallographic groups
1596:
1429:(2,3,7) triangle group
1411:
1301:
951:
921:
885:
860:
840:
807:
779:
757:
737:
700:
680:
656:
632:
579:
559:
535:
507:
484:
465:orientation-preserving
450:
354:Möbius transformations
342:
308:
229:
205:
104:non-Euclidean geometry
1650:J. Reine Angew. Math.
1597:
1412:
1302:
1166:is said to be of the
952:
922:
886:
861:
841:
808:
780:
758:
738:
701:
681:
657:
633:
580:
560:
536:
508:
485:
451:
343:
309:
230:
206:
1735:, and David Wright,
1619:Quasi-Fuchsian group
1542:
1368:
1217:
981:) will preserve the
935:
927:: elements of PSL(2,
899:
875:
850:
817:
789:
767:
747:
720:
712:, which means that:
690:
670:
646:
592:
569:
549:
525:
497:
474:
363:
332:
246:
219:
153:
1849:Hyperbolic geometry
1755:Peter J. Nicholls,
1660:Hershel M. Farkas,
1439:, as well as other
1349:) for each integer
1122:accumulation points
950:{\displaystyle z=0}
1814:Vinberg, Ernest B.
1790:10.1007/BF02592124
1592:
1407:
1401:
1297:
1250:
973:General definition
947:
917:
881:
856:
836:
803:
785:such that for all
775:
753:
733:
696:
676:
652:
640:accumulation point
628:
575:
555:
531:
503:
480:
446:
396:
338:
304:
235:is a model of the
225:
201:
127:Henri Poincaré
112:Euclidean geometry
65:finitely generated
39:. The group PSL(2,
1765:978-0-521-37674-7
1747:978-0-521-35253-6
1724:978-0-226-42583-2
1704:978-0-8218-3160-1
1681:(See section 1.6)
1678:978-0-8218-1392-8
1672:, Providence RI,
1587:
1515:Metric properties
1295:
884:{\displaystyle H}
859:{\displaystyle I}
655:{\displaystyle H}
578:{\displaystyle H}
558:{\displaystyle z}
534:{\displaystyle H}
493:A Fuchsian group
483:{\displaystyle H}
441:
341:{\displaystyle H}
299:
266:
228:{\displaystyle H}
29:discrete subgroup
1871:
1854:Riemann surfaces
1830:
1818:"Fuchsian group"
1809:
1792:
1776:Acta Mathematica
1707:(See Chapter 2.)
1657:
1601:
1599:
1598:
1593:
1588:
1580:
1566:
1557:
1549:
1437:Macbeath surface
1416:
1414:
1413:
1408:
1406:
1405:
1353:> 0. Here Γ(
1306:
1304:
1303:
1298:
1296:
1294:
1280:
1266:
1255:
1254:
1102:upper half-plane
1087:rational numbers
1015:. Let Γ ⊂ PSL(2,
1008:) preserving Δ.
956:
954:
953:
948:
926:
924:
923:
918:
890:
888:
887:
882:
865:
863:
862:
857:
845:
843:
842:
837:
829:
828:
812:
810:
809:
804:
802:
784:
782:
781:
776:
774:
762:
760:
759:
754:
742:
740:
739:
734:
732:
731:
705:
703:
702:
697:
685:
683:
682:
677:
661:
659:
658:
653:
637:
635:
634:
629:
584:
582:
581:
576:
564:
562:
561:
556:
540:
538:
537:
532:
512:
510:
509:
504:
489:
487:
486:
481:
455:
453:
452:
447:
442:
440:
426:
412:
401:
400:
347:
345:
344:
339:
313:
311:
310:
305:
300:
298:
297:
282:
281:
269:
267:
259:
237:hyperbolic plane
234:
232:
231:
226:
213:upper half-plane
210:
208:
207:
202:
191:
180:
175:
96:Riemann surfaces
61:upper half plane
53:hyperbolic plane
1879:
1878:
1874:
1873:
1872:
1870:
1869:
1868:
1859:Discrete groups
1844:Kleinian groups
1834:
1833:
1771:Poincaré, Henri
1733:Caroline Series
1716:Fuchsian Groups
1637:
1615:
1579:
1562:
1550:
1545:
1543:
1540:
1539:
1517:
1474:) are Fuchsian.
1400:
1399:
1394:
1388:
1387:
1382:
1372:
1371:
1369:
1366:
1365:
1340:elliptic curves
1281:
1267:
1265:
1249:
1248:
1243:
1237:
1236:
1231:
1221:
1220:
1218:
1215:
1214:
1196:
1110:
975:
936:
933:
932:
900:
897:
896:
876:
873:
872:
851:
848:
847:
824:
820:
818:
815:
814:
798:
790:
787:
786:
770:
768:
765:
764:
748:
745:
744:
743:of elements of
727:
723:
721:
718:
717:
716:Every sequence
691:
688:
687:
671:
668:
667:
647:
644:
643:
593:
590:
589:
570:
567:
566:
550:
547:
546:
526:
523:
522:
519:discontinuously
498:
495:
494:
475:
472:
471:
427:
413:
411:
395:
394:
389:
383:
382:
377:
367:
366:
364:
361:
360:
352:(also known as
333:
330:
329:
293:
289:
277:
273:
268:
258:
247:
244:
243:
220:
217:
216:
187:
176:
171:
154:
151:
150:
147:
92:Fuchsian models
17:
12:
11:
5:
1877:
1867:
1866:
1861:
1856:
1851:
1846:
1832:
1831:
1810:
1767:
1753:
1726:
1712:Svetlana Katok
1709:
1686:Henryk Iwaniec
1683:
1658:
1642:Fuchs, Lazarus
1636:
1633:
1632:
1631:
1629:Schottky group
1626:
1621:
1614:
1611:
1603:
1602:
1591:
1586:
1583:
1578:
1575:
1572:
1569:
1565:
1561:
1556:
1553:
1548:
1516:
1513:
1512:
1511:
1510:) is Fuchsian.
1506:of Γ in PSL(2,
1500:
1489:
1482:
1475:
1454:All these are
1449:triangle group
1445:von Dyck group
1441:Hurwitz groups
1418:
1417:
1404:
1398:
1395:
1393:
1390:
1389:
1386:
1383:
1381:
1378:
1377:
1375:
1357:) consists of
1308:
1307:
1293:
1290:
1287:
1284:
1279:
1276:
1273:
1270:
1264:
1261:
1258:
1253:
1247:
1244:
1242:
1239:
1238:
1235:
1232:
1230:
1227:
1226:
1224:
1195:
1192:
1164:Fuchsian group
1140:. Then Λ(Γ) ⊆
1109:
1106:
1098:open unit disk
1091:Kleinian group
1063:
1062:
1055:
1050:at each point
1044:
1037:discrete group
1013:Fuchsian group
983:Riemann sphere
974:
971:
946:
943:
940:
916:
913:
910:
907:
904:
880:
868:
867:
855:
835:
832:
827:
823:
801:
797:
794:
773:
752:
730:
726:
709:discrete group
695:
675:
664:
663:
651:
627:
624:
621:
618:
615:
612:
609:
606:
603:
600:
597:
574:
554:
530:
517:), which acts
502:
479:
457:
456:
445:
439:
436:
433:
430:
425:
422:
419:
416:
410:
407:
404:
399:
393:
390:
388:
385:
384:
381:
378:
376:
373:
372:
370:
337:
315:
314:
303:
296:
292:
288:
285:
280:
276:
272:
265:
262:
257:
254:
251:
224:
200:
197:
194:
190:
186:
183:
179:
174:
170:
167:
164:
161:
158:
146:
143:
73:Kleinian group
25:Fuchsian group
15:
9:
6:
4:
3:
2:
1876:
1865:
1862:
1860:
1857:
1855:
1852:
1850:
1847:
1845:
1842:
1841:
1839:
1829:
1825:
1824:
1819:
1815:
1811:
1808:
1804:
1800:
1796:
1791:
1786:
1782:
1778:
1777:
1772:
1768:
1766:
1762:
1758:
1754:
1752:
1748:
1744:
1740:
1739:
1734:
1730:
1729:David Mumford
1727:
1725:
1721:
1717:
1713:
1710:
1708:
1705:
1701:
1697:
1696:
1691:
1687:
1684:
1682:
1679:
1675:
1671:
1667:
1663:
1659:
1655:
1651:
1647:
1643:
1639:
1638:
1630:
1627:
1625:
1622:
1620:
1617:
1616:
1610:
1608:
1589:
1584:
1581:
1576:
1573:
1570:
1567:
1559:
1538:
1537:
1536:
1534:
1530:
1526:
1522:
1509:
1505:
1501:
1498:
1494:
1490:
1487:
1483:
1480:
1476:
1473:
1469:
1465:
1461:
1460:
1459:
1457:
1452:
1450:
1446:
1442:
1438:
1434:
1433:Klein quartic
1430:
1425:
1423:
1402:
1396:
1391:
1384:
1379:
1373:
1364:
1363:
1362:
1360:
1356:
1352:
1348:
1343:
1341:
1337:
1333:
1329:
1325:
1321:
1317:
1313:
1291:
1288:
1285:
1282:
1277:
1274:
1271:
1268:
1262:
1259:
1256:
1251:
1245:
1240:
1233:
1228:
1222:
1213:
1212:
1211:
1209:
1205:
1201:
1200:modular group
1191:
1187:
1185:
1181:
1177:
1176:nowhere dense
1173:
1169:
1165:
1162:Otherwise, a
1160:
1158:
1154:
1150:
1145:
1143:
1139:
1135:
1131:
1127:
1123:
1119:
1115:
1105:
1103:
1099:
1094:
1092:
1088:
1084:
1080:
1076:
1075:modular group
1072:
1068:
1060:
1056:
1053:
1049:
1045:
1042:
1038:
1034:
1033:
1032:
1030:
1026:
1022:
1018:
1014:
1009:
1007:
1003:
999:
995:
991:
987:
984:
980:
970:
968:
964:
960:
944:
941:
938:
931:) will carry
930:
914:
911:
908:
905:
902:
894:
878:
853:
833:
830:
825:
821:
795:
792:
728:
724:
715:
714:
713:
711:
710:
649:
641:
619:
616:
613:
610:
607:
601:
598:
588:
572:
552:
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1168:second type
1116:of a point
21:mathematics
1838:Categories
1807:14.0338.01
1635:References
1504:normalizer
1464:hyperbolic
1184:Cantor set
1108:Limit sets
1061:Ω(Γ) of Γ.
545:For every
468:isometries
317:The group
135:Fuchs 1880
120:disc model
49:isometries
1828:EMS Press
1816:(2001) ,
1799:0001-5962
1662:Irwin Kra
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1864:Fractals
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1613:See also
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1194:Examples
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1029:Fuchsian
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