1937:
2293:
997:
3595:
2429:
2125:
2034:
1224:
434:
2640:
2533:
892:
3163:
3073:
2358:
1766:
1660:
701:
342:
2692:
2578:
2474:
584:
2080:
1989:
2989:
284:
3111:
2319:
2154:
1511:
3946:
3893:
3844:
3755:
3703:
3654:
3521:
3466:
466:
146:
115:
3406:
767:
4397:
4107:
653:
3293:
is a uniform convergence action (see for a formal proof). Bowditch proved an important converse, thus obtaining a topological characterization of word-hyperbolic groups:
1866:
617:
3291:
2870:
2817:
1436:
1326:
1277:
1141:
1026:
3021:
733:
374:
3197:
2898:
2841:
2788:
2764:
2720:
2054:
1963:
1839:
1819:
1786:
1727:
1707:
1680:
1621:
1601:
1574:
1554:
1534:
1465:
1413:
1373:
1177:
1112:
1072:
795:
243:
195:
45:
2944:
2206:
2180:
3991:
3917:
3864:
3799:
3775:
3726:
3674:
3625:
3541:
3492:
3434:
3377:
3357:
3337:
3317:
3268:
3237:
3217:
2918:
2740:
1485:
1393:
1349:
1303:
1254:
1092:
1046:
835:
815:
546:
526:
506:
486:
215:
78:
4143:
4078:
1871:
2211:
3971:, originally defined in the context of Kleinian and word-hyperbolic groups, can be defined and studied in the context of setting of convergence groups.
897:
4223:
3546:
2363:
4426:
4194:
4057:
3961:
in terms of convergence actions, generalizing
Bowditch's characterization of word-hyperbolic groups as uniform convergence groups.
2085:
1994:
1182:
379:
4521:
4172:
2583:
2479:
840:
1144:
3964:
One can consider more general versions of group actions with "convergence property" without the discreteness assumption.
3116:
3026:
2324:
1732:
1626:
4516:
3958:
1283:
4511:
657:
47:
289:
1094:
with at least two points. Then this action is a discrete convergence action if and only if the induced action of
2645:
4296:
2538:
2434:
551:
3968:
1227:
85:
2059:
1968:
777:
The above definition of convergence group admits a useful equivalent reformulation in terms of the action of
2961:
256:
4179:. Mathematical Sciences Research Institute Publications. Vol. 8. New York: Springer. pp. 75–263.
3078:
3896:
3810:
3469:
2820:
166:
162:
4506:
4366:
4259:
3248:
2298:
2133:
1490:
1234:
1352:
3922:
3869:
3820:
3731:
3679:
3630:
3497:
3442:
439:
122:
91:
4305:
3801:
contains a finite index subgroup isomorphic to the fundamental group of a closed hyperbolic surface.
3382:
738:
25:
622:
4470:
4276:
1844:
589:
158:
4294:
Casson, Andrew; Jungreis, Douglas (1994). "Convergence groups and
Seifert fibered 3-manifolds".
3273:
2846:
2793:
1418:
1308:
1259:
1117:
1002:
4190:
4053:
2994:
1513:. Then it is known (Lemma 3.1 in or Lemma 6.2 in ) that exactly one of the following occurs:
706:
347:
150:
3182:
2883:
2826:
2773:
2749:
2705:
2039:
1942:
1824:
1798:
1771:
1712:
1692:
1665:
1606:
1586:
1559:
1539:
1519:
1450:
1398:
1358:
1162:
1097:
1057:
780:
228:
180:
30:
4480:
4435:
4406:
4313:
4268:
4232:
4180:
4152:
4116:
4087:
4045:
4008:
4000:
3437:
2923:
2185:
2159:
118:
58:
4204:
4200:
3814:
3656:
then this action is topologically conjugate to an action induced by a geometric action of
4309:
4073:
4037:
3902:
3849:
3784:
3760:
3711:
3659:
3610:
3526:
3477:
3419:
3362:
3342:
3322:
3302:
3253:
3222:
3202:
3173:
2903:
2725:
1932:{\displaystyle \operatorname {Fix} _{M}(\gamma )=\operatorname {Fix} _{M}(\gamma ^{p})}
1470:
1378:
1334:
1288:
1239:
1157:
1077:
1031:
820:
800:
531:
511:
491:
488:. Here converging uniformly on compact subsets means that for every open neighborhood
471:
200:
81:
63:
4395:
Yaman, Asli (2004). "A topological characterisation of relatively hyperbolic groups".
4332:
3601:(1992), Casson–Jungreis (1994), and Freden (1995) shows that the converse also holds:
4500:
4454:
55:
51:
4424:
Gerasimov, Victor (2009). "Expansive convergence groups are relatively hyperbolic".
2288:{\displaystyle \lim _{n\to \infty }\gamma ^{n}x=\lim _{n\to -\infty }\gamma ^{n}x=a}
154:
4254:
4157:
4138:
3474:
if this action is properly discontinuous and cocompact. Every geometric action of
4250:
4218:
4185:
3598:
3169:
4004:
3247:
It was already observed by Gromov that the natural action by translations of a
4439:
4049:
4044:. De Gruyter Proceedings in Mathematics. de Gruyter, Berlin. pp. 23–54.
4105:
Tukia, Pekka (1998). "Conical limit points and uniform convergence groups".
4076:(1999). "Treelike structures arising from continua and convergence groups".
3778:
4410:
4120:
4013:
3989:
Gehring, F. W.; Martin, G. J. (1987). "Discrete quasiconformal groups I".
3319:
act as a discrete uniform convergence group on a compact metrizable space
4374:
Ergodic theory, symbolic dynamics, and hyperbolic spaces (Trieste, 1989)
4091:
4485:
4458:
4317:
4280:
4236:
3895:
then this action is topologically conjugate to an action induced by a
992:{\displaystyle \Delta (M)=\{(a,b,c)\in M^{3}\mid \#\{a,b,c\}\leq 2\}}
253:
for this action) if for every infinite distinct sequence of elements
4272:
4376:. Oxford Sci. Publ., Oxford Univ. Press, New York. pp. 315–369
3817:
in terms of word-hyperbolic groups with boundaries homeomorphic to
4475:
1467:
be a group acting by homeomorphisms on a compact metrizable space
1074:
be a group acting by homeomorphisms on a compact metrizable space
436:
converge uniformly on compact subsets to the constant map sending
197:
be a group acting by homeomorphisms on a compact metrizable space
3590:{\displaystyle \mathbb {S} ^{1}=\partial H^{2}\approx \partial G}
2424:{\displaystyle \operatorname {Fix} _{M}(\gamma )=\{a_{-},a_{+}\}}
3866:
is a group acting as a discrete uniform convergence group on
3627:
is a group acting as a discrete uniform convergence group on
2843:
is a uniform convergence group if and only if its action on
2642:, and these convergences are uniform on compact subsets of
2120:{\displaystyle M\setminus \operatorname {Fix} _{M}(\gamma )}
2029:{\displaystyle M\setminus \operatorname {Fix} _{M}(\gamma )}
1375:
be a group acting properly discontinuously by isometries on
2958:) if there exist an infinite sequence of distinct elements
4333:"Negatively curved groups have the convergence property I"
1219:{\displaystyle \mathbb {S} ^{2}=\partial \mathbb {H} ^{3}}
429:{\displaystyle \gamma _{n_{k}}{\big |}_{M\setminus \{a\}}}
80:
in a way that generalizes the properties of the action of
4175:(1987). "Hyperbolic groups". In Gersten, Steve M. (ed.).
773:
Reformulation in terms of the action on distinct triples
4040:(1999). "Convergence groups and configuration spaces".
2635:{\displaystyle \lim _{n\to -\infty }\gamma ^{n}x=a_{-}}
3219:
is uniform if and only if every non-isolated point of
2528:{\displaystyle \lim _{n\to \infty }\gamma ^{n}x=a_{+}}
149:. The notion of a convergence group was introduced by
4139:"A topological characterisation of hyperbolic groups"
3925:
3905:
3872:
3852:
3823:
3787:
3763:
3734:
3714:
3682:
3662:
3633:
3613:
3549:
3529:
3500:
3480:
3445:
3422:
3385:
3365:
3345:
3325:
3305:
3276:
3256:
3225:
3205:
3185:
3119:
3081:
3029:
2997:
2964:
2926:
2906:
2886:
2849:
2829:
2796:
2776:
2752:
2728:
2708:
2648:
2586:
2541:
2482:
2437:
2366:
2327:
2301:
2214:
2188:
2162:
2136:
2088:
2062:
2042:
1997:
1971:
1945:
1874:
1847:
1827:
1801:
1774:
1735:
1715:
1695:
1668:
1629:
1609:
1589:
1562:
1542:
1522:
1493:
1473:
1453:
1421:
1401:
1381:
1361:
1337:
1311:
1291:
1262:
1242:
1185:
1165:
1120:
1100:
1080:
1060:
1034:
1005:
900:
887:{\displaystyle \Theta (M):=M^{3}\setminus \Delta (M)}
843:
823:
803:
783:
741:
709:
660:
625:
592:
554:
534:
514:
494:
474:
442:
382:
350:
292:
259:
231:
203:
183:
125:
94:
66:
33:
4457:; Leininger, Christopher; Ohshika, Ken'ichi (2016).
4367:"The theory of negatively curved spaces and groups"
3948:by isometries. This conjecture still remains open.
4459:"Conical limit points and the Cannon-Thurston map"
4042:Geometric group theory down under (Canberra, 1996)
3940:
3911:
3887:
3858:
3838:
3793:
3769:
3749:
3720:
3697:
3668:
3648:
3619:
3589:
3535:
3515:
3486:
3460:
3428:
3400:
3371:
3351:
3331:
3311:
3285:
3262:
3231:
3211:
3191:
3158:{\displaystyle \lim _{n\to \infty }\gamma _{n}y=b}
3157:
3105:
3068:{\displaystyle \lim _{n\to \infty }\gamma _{n}x=a}
3067:
3015:
2983:
2938:
2912:
2892:
2864:
2835:
2811:
2782:
2758:
2734:
2714:
2686:
2634:
2572:
2527:
2468:
2423:
2352:
2313:
2287:
2200:
2174:
2148:
2119:
2074:
2048:
2028:
1983:
1957:
1931:
1860:
1833:
1813:
1780:
1760:
1721:
1701:
1674:
1654:
1615:
1595:
1568:
1548:
1528:
1505:
1479:
1459:
1438:is a discrete convergence action (Lemma 2.11 of ).
1430:
1407:
1387:
1367:
1343:
1320:
1297:
1271:
1248:
1218:
1171:
1135:
1106:
1086:
1066:
1040:
1020:
991:
886:
829:
809:
789:
761:
727:
695:
647:
611:
578:
540:
520:
500:
480:
460:
428:
368:
336:
278:
237:
209:
189:
140:
109:
72:
39:
2353:{\displaystyle \operatorname {Fix} _{M}(\gamma )}
2082:acts properly discontinuously and cocompactly on
1761:{\displaystyle \operatorname {Fix} _{M}(\gamma )}
1655:{\displaystyle \operatorname {Fix} _{M}(\gamma )}
1051:Then the following equivalence is known to hold:
157:(1987) and has since found wide applications in
3121:
3031:
2588:
2484:
2248:
2216:
1443:Classification of elements in convergence groups
4398:Journal fĂĽr die reine und angewandte Mathematik
4108:Journal fĂĽr die reine und angewandte Mathematik
3179:A discrete convergence group action of a group
2872:is both properly discontinuous and co-compact.
1868:have the same type. Also in cases (2) and (3)
3992:Proceedings of the London Mathematical Society
1768:consists of two distinct points; in this case
1028:is called the "space of distinct triples" for
696:{\displaystyle \gamma _{n_{k}}(K)\subseteq U}
403:
8:
4144:Journal of the American Mathematical Society
4079:Memoirs of the American Mathematical Society
4032:
4030:
4028:
4026:
4024:
3100:
3094:
2681:
2655:
2567:
2554:
2463:
2450:
2418:
2392:
2069:
2063:
1978:
1972:
1395:. Then the corresponding boundary action of
986:
977:
959:
916:
573:
567:
455:
449:
421:
415:
337:{\displaystyle \gamma _{n_{k}},k=1,2,\dots }
4132:
4130:
3243:Word-hyperbolic groups and their boundaries
3967:The most general version of the notion of
2687:{\displaystyle M\setminus \{a_{-},a_{+}\}}
4484:
4474:
4184:
4156:
4012:
3932:
3928:
3927:
3924:
3904:
3879:
3875:
3874:
3871:
3851:
3830:
3826:
3825:
3822:
3786:
3762:
3741:
3737:
3736:
3733:
3713:
3689:
3685:
3684:
3681:
3661:
3640:
3636:
3635:
3632:
3612:
3572:
3556:
3552:
3551:
3548:
3528:
3507:
3503:
3502:
3499:
3479:
3452:
3448:
3447:
3444:
3421:
3384:
3364:
3344:
3324:
3304:
3275:
3255:
3224:
3204:
3184:
3140:
3124:
3118:
3080:
3050:
3034:
3028:
2996:
2969:
2963:
2925:
2920:as a discrete convergence group. A point
2905:
2885:
2848:
2828:
2795:
2775:
2751:
2727:
2707:
2702:A discrete convergence action of a group
2675:
2662:
2647:
2626:
2610:
2591:
2585:
2573:{\displaystyle x\in M\setminus \{a_{+}\}}
2561:
2540:
2519:
2503:
2487:
2481:
2469:{\displaystyle x\in M\setminus \{a_{-}\}}
2457:
2436:
2412:
2399:
2371:
2365:
2332:
2326:
2300:
2270:
2251:
2235:
2219:
2213:
2187:
2161:
2135:
2099:
2087:
2061:
2041:
2008:
1996:
1970:
1944:
1920:
1904:
1879:
1873:
1852:
1846:
1826:
1800:
1773:
1740:
1734:
1714:
1694:
1667:
1634:
1628:
1608:
1588:
1561:
1541:
1521:
1492:
1472:
1452:
1420:
1400:
1380:
1360:
1336:
1310:
1305:by translations on its Bowditch boundary
1290:
1261:
1241:
1210:
1206:
1205:
1192:
1188:
1187:
1184:
1164:
1119:
1099:
1079:
1059:
1033:
1004:
947:
899:
863:
842:
822:
802:
782:
751:
746:
740:
708:
670:
665:
659:
636:
624:
597:
591:
579:{\displaystyle K\subset M\setminus \{a\}}
553:
533:
513:
493:
473:
441:
408:
402:
401:
392:
387:
381:
349:
302:
297:
291:
264:
258:
230:
202:
182:
132:
128:
127:
124:
101:
97:
96:
93:
65:
32:
4255:"Convergence groups are Fuchsian groups"
3952:Applications and further generalizations
3809:One of the equivalent reformulations of
3523:induces a uniform convergence action of
3339:with no isolated points. Then the group
3981:
3597:. An important result of Tukia (1986),
3091:
2652:
2551:
2447:
2092:
2075:{\displaystyle \langle \gamma \rangle }
2001:
1984:{\displaystyle \langle \gamma \rangle }
869:
564:
446:
412:
4340:Annales Academiae Scientiarum Fennicae
3781:a hyperbolic surface group, that is,
3359:is word-hyperbolic and there exists a
2984:{\displaystyle \gamma _{n}\in \Gamma }
1256:by translations on its ideal boundary
797:on the "space of distinct triples" of
279:{\displaystyle \gamma _{n}\in \Gamma }
3106:{\displaystyle y\in M\setminus \{x\}}
1487:with at least three points, and let
7:
4221:(1986). "On quasiconformal groups".
3805:Convergence actions on the 2-sphere
3581:
3565:
3392:
3277:
3186:
3131:
3041:
2978:
2900:act on a compact metrizable space
2887:
2850:
2830:
2797:
2777:
2753:
2709:
2601:
2494:
2314:{\displaystyle \gamma \in \Gamma }
2308:
2261:
2226:
2149:{\displaystyle \gamma \in \Gamma }
2143:
1716:
1610:
1543:
1506:{\displaystyle \gamma \in \Gamma }
1500:
1454:
1422:
1402:
1362:
1312:
1263:
1201:
1166:
1121:
1101:
1061:
1006:
956:
901:
872:
844:
784:
769:are not required to be distinct.
273:
232:
184:
34:
14:
4427:Geometric and Functional Analysis
3957:Yaman gave a characterization of
3412:Convergence actions on the circle
3172:, also independently obtained by
1991:acts properly discontinuously on
735:associated with the subsequence
3941:{\displaystyle \mathbb {H} ^{3}}
3888:{\displaystyle \mathbb {S} ^{2}}
3839:{\displaystyle \mathbb {S} ^{2}}
3750:{\displaystyle \mathbb {H} ^{2}}
3698:{\displaystyle \mathbb {H} ^{2}}
3649:{\displaystyle \mathbb {S} ^{1}}
3516:{\displaystyle \mathbb {H} ^{2}}
3461:{\displaystyle \mathbb {H} ^{2}}
2156:is parabolic with a fixed point
1662:is a single point; in this case
461:{\displaystyle M\setminus \{a\}}
141:{\displaystyle \mathbb {H} ^{3}}
110:{\displaystyle \mathbb {S} ^{2}}
4463:Conformal Geometry and Dynamics
3416:An isometric action of a group
3401:{\displaystyle M\to \partial G}
762:{\displaystyle \gamma _{n_{k}}}
4224:Journal d'Analyse Mathématique
3389:
3199:on a compact metrizable space
3128:
3038:
2859:
2853:
2806:
2800:
2722:on a compact metrizable space
2595:
2491:
2386:
2380:
2347:
2341:
2255:
2223:
2114:
2108:
2023:
2017:
1926:
1913:
1894:
1888:
1755:
1749:
1649:
1643:
1328:is a convergence group action.
1279:is a convergence group action.
1230:is a convergence group action.
1130:
1124:
1015:
1009:
937:
919:
910:
904:
881:
875:
853:
847:
684:
678:
1:
4158:10.1090/S0894-0347-98-00264-1
3959:relatively hyperbolic groups
648:{\displaystyle k\geq k_{0},}
4186:10.1007/978-1-4613-9586-7_3
4137:Bowditch, Brian H. (1998).
3379:-equivariant homeomorphism
1861:{\displaystyle \gamma ^{p}}
1284:relatively hyperbolic group
612:{\displaystyle k_{0}\geq 1}
223:discrete convergence action
4538:
3286:{\displaystyle \partial G}
3239:is a conical limit point.
2865:{\displaystyle \Theta (M)}
2812:{\displaystyle \Theta (M)}
2698:Uniform convergence groups
1431:{\displaystyle \partial X}
1321:{\displaystyle \partial G}
1272:{\displaystyle \partial G}
1136:{\displaystyle \Theta (M)}
1021:{\displaystyle \Theta (M)}
286:there exist a subsequence
251:discrete convergence group
217:. This action is called a
22:discrete convergence group
4440:10.1007/s00039-009-0718-7
4365:Cannon, James W. (1991).
2950:(sometimes also called a
2768:uniform convergence group
4331:Freden, Eric M. (1995).
4297:Inventiones Mathematicae
4050:10.1515/9783110806861.23
4005:10.1093/plms/s3-55_2.331
3016:{\displaystyle a,b\in M}
728:{\displaystyle a,b\in M}
703:. Note that the "poles"
369:{\displaystyle a,b\in M}
3192:{\displaystyle \Gamma }
3168:An important result of
2893:{\displaystyle \Gamma }
2836:{\displaystyle \Gamma }
2783:{\displaystyle \Gamma }
2759:{\displaystyle \Gamma }
2715:{\displaystyle \Gamma }
2049:{\displaystyle \gamma }
1958:{\displaystyle p\neq 0}
1834:{\displaystyle \gamma }
1814:{\displaystyle p\neq 0}
1781:{\displaystyle \gamma }
1722:{\displaystyle \Gamma }
1702:{\displaystyle \gamma }
1675:{\displaystyle \gamma }
1616:{\displaystyle \Gamma }
1596:{\displaystyle \gamma }
1569:{\displaystyle \gamma }
1549:{\displaystyle \Gamma }
1529:{\displaystyle \gamma }
1460:{\displaystyle \Gamma }
1408:{\displaystyle \Gamma }
1368:{\displaystyle \Gamma }
1172:{\displaystyle \Gamma }
1107:{\displaystyle \Gamma }
1067:{\displaystyle \Gamma }
790:{\displaystyle \Gamma }
238:{\displaystyle \Gamma }
190:{\displaystyle \Gamma }
163:quasiconformal analysis
40:{\displaystyle \Gamma }
4522:Geometric group theory
4177:Essays in group theory
3942:
3913:
3889:
3860:
3840:
3813:, originally posed by
3795:
3771:
3751:
3728:acts geometrically on
3722:
3699:
3670:
3650:
3621:
3591:
3537:
3517:
3488:
3462:
3430:
3402:
3373:
3353:
3333:
3313:
3287:
3264:
3233:
3213:
3193:
3159:
3107:
3069:
3017:
2985:
2956:point of approximation
2940:
2939:{\displaystyle x\in M}
2914:
2894:
2866:
2837:
2813:
2784:
2760:
2736:
2716:
2688:
2636:
2574:
2529:
2470:
2425:
2354:
2315:
2289:
2202:
2201:{\displaystyle x\in M}
2176:
2175:{\displaystyle a\in M}
2150:
2121:
2076:
2050:
2030:
1985:
1959:
1933:
1862:
1835:
1815:
1782:
1762:
1723:
1709:has infinite order in
1703:
1676:
1656:
1617:
1603:has infinite order in
1597:
1570:
1550:
1530:
1507:
1481:
1461:
1432:
1409:
1389:
1369:
1345:
1322:
1299:
1273:
1250:
1228:Möbius transformations
1220:
1173:
1145:properly discontinuous
1137:
1108:
1088:
1068:
1042:
1022:
993:
888:
831:
811:
791:
763:
729:
697:
649:
613:
586:there exists an index
580:
542:
522:
502:
482:
462:
430:
370:
338:
280:
239:
211:
191:
167:geometric group theory
142:
111:
88:on the ideal boundary
86:Möbius transformations
74:
41:
4411:10.1515/crll.2004.007
4260:Annals of Mathematics
4121:10.1515/crll.1998.081
3943:
3914:
3890:
3861:
3841:
3796:
3772:
3752:
3723:
3700:
3671:
3651:
3622:
3592:
3538:
3518:
3489:
3463:
3431:
3403:
3374:
3354:
3334:
3314:
3288:
3265:
3249:word-hyperbolic group
3234:
3214:
3194:
3160:
3108:
3070:
3018:
2986:
2941:
2915:
2895:
2867:
2838:
2814:
2785:
2761:
2737:
2717:
2689:
2637:
2575:
2530:
2471:
2426:
2355:
2316:
2290:
2203:
2177:
2151:
2122:
2077:
2051:
2031:
1986:
1960:
1934:
1863:
1836:
1816:
1783:
1763:
1724:
1704:
1677:
1657:
1618:
1598:
1571:
1551:
1531:
1508:
1482:
1462:
1433:
1410:
1390:
1370:
1355:metric space and let
1351:be a proper geodesic
1346:
1323:
1300:
1274:
1251:
1235:word-hyperbolic group
1221:
1174:
1138:
1109:
1089:
1069:
1043:
1023:
994:
889:
832:
812:
792:
764:
730:
698:
650:
614:
581:
543:
523:
503:
483:
463:
431:
371:
339:
281:
240:
212:
192:
143:
112:
75:
42:
3923:
3903:
3870:
3850:
3821:
3785:
3761:
3732:
3712:
3680:
3660:
3631:
3611:
3547:
3527:
3498:
3478:
3443:
3420:
3383:
3363:
3343:
3323:
3303:
3274:
3254:
3223:
3203:
3183:
3117:
3079:
3027:
2995:
2991:and distinct points
2962:
2924:
2904:
2884:
2876:Conical limit points
2847:
2827:
2794:
2774:
2750:
2726:
2706:
2646:
2584:
2539:
2480:
2435:
2364:
2325:
2321:is loxodromic, then
2299:
2212:
2186:
2160:
2134:
2086:
2060:
2056:is loxodromic, then
2040:
1995:
1969:
1943:
1872:
1845:
1825:
1799:
1795:Moreover, for every
1772:
1733:
1713:
1693:
1666:
1627:
1607:
1587:
1560:
1540:
1536:has finite order in
1520:
1491:
1471:
1451:
1419:
1399:
1379:
1359:
1335:
1309:
1289:
1260:
1240:
1183:
1163:
1118:
1098:
1078:
1058:
1032:
1003:
898:
841:
821:
801:
781:
739:
707:
658:
623:
619:such that for every
590:
552:
532:
512:
492:
472:
440:
380:
348:
290:
257:
229:
201:
181:
123:
92:
64:
31:
4310:1994InMat.118..441C
3969:Cannon–Thurston map
3811:Cannon's conjecture
3708:Note that whenever
2948:conical limit point
2770:) if the action of
2036:. Additionally, if
376:such that the maps
119:hyperbolic 3-space
4517:Geometric topology
4318:10.1007/BF01231540
4237:10.1007/BF02796595
3938:
3909:
3885:
3856:
3836:
3791:
3767:
3747:
3718:
3695:
3666:
3646:
3617:
3587:
3533:
3513:
3484:
3458:
3426:
3398:
3369:
3349:
3329:
3309:
3283:
3260:
3229:
3209:
3189:
3155:
3135:
3103:
3065:
3045:
3013:
2981:
2952:radial limit point
2936:
2910:
2890:
2862:
2833:
2809:
2780:
2756:
2732:
2712:
2684:
2632:
2605:
2570:
2525:
2498:
2466:
2431:so that for every
2421:
2360:can be written as
2350:
2311:
2285:
2265:
2230:
2198:
2172:
2146:
2117:
2072:
2046:
2026:
1981:
1955:
1929:
1858:
1831:
1811:
1778:
1758:
1729:and the fixed set
1719:
1699:
1672:
1652:
1623:and the fixed set
1613:
1593:
1566:
1546:
1526:
1503:
1477:
1457:
1428:
1405:
1385:
1365:
1341:
1318:
1295:
1269:
1246:
1216:
1169:
1133:
1104:
1084:
1064:
1038:
1018:
989:
884:
827:
807:
787:
759:
725:
693:
645:
609:
576:
548:and every compact
538:
518:
498:
478:
458:
426:
366:
334:
276:
235:
219:convergence action
207:
187:
159:geometric topology
138:
107:
70:
37:
16:In mathematics, a
4512:Dynamical systems
4263:. Second series.
4092:10.1090/memo/0662
3912:{\displaystyle G}
3859:{\displaystyle G}
3794:{\displaystyle G}
3770:{\displaystyle G}
3721:{\displaystyle G}
3669:{\displaystyle G}
3620:{\displaystyle G}
3536:{\displaystyle G}
3487:{\displaystyle G}
3429:{\displaystyle G}
3372:{\displaystyle G}
3352:{\displaystyle G}
3332:{\displaystyle M}
3312:{\displaystyle G}
3263:{\displaystyle G}
3232:{\displaystyle M}
3212:{\displaystyle M}
3120:
3030:
2913:{\displaystyle M}
2735:{\displaystyle M}
2587:
2483:
2247:
2215:
1480:{\displaystyle M}
1388:{\displaystyle X}
1353:Gromov-hyperbolic
1344:{\displaystyle X}
1298:{\displaystyle G}
1249:{\displaystyle G}
1087:{\displaystyle M}
1041:{\displaystyle M}
830:{\displaystyle M}
810:{\displaystyle M}
541:{\displaystyle M}
521:{\displaystyle b}
501:{\displaystyle U}
481:{\displaystyle b}
247:convergence group
210:{\displaystyle M}
173:Formal definition
73:{\displaystyle M}
18:convergence group
4529:
4491:
4490:
4488:
4486:10.1090/ecgd/294
4478:
4450:
4444:
4443:
4421:
4415:
4414:
4392:
4386:
4385:
4383:
4381:
4371:
4362:
4356:
4355:
4353:
4351:
4337:
4328:
4322:
4321:
4291:
4285:
4284:
4247:
4241:
4240:
4215:
4209:
4208:
4188:
4169:
4163:
4162:
4160:
4134:
4125:
4124:
4102:
4096:
4095:
4070:
4064:
4063:
4034:
4019:
4018:
4016:
3986:
3947:
3945:
3944:
3939:
3937:
3936:
3931:
3918:
3916:
3915:
3910:
3897:geometric action
3894:
3892:
3891:
3886:
3884:
3883:
3878:
3865:
3863:
3862:
3857:
3845:
3843:
3842:
3837:
3835:
3834:
3829:
3800:
3798:
3797:
3792:
3776:
3774:
3773:
3768:
3756:
3754:
3753:
3748:
3746:
3745:
3740:
3727:
3725:
3724:
3719:
3704:
3702:
3701:
3696:
3694:
3693:
3688:
3675:
3673:
3672:
3667:
3655:
3653:
3652:
3647:
3645:
3644:
3639:
3626:
3624:
3623:
3618:
3596:
3594:
3593:
3588:
3577:
3576:
3561:
3560:
3555:
3542:
3540:
3539:
3534:
3522:
3520:
3519:
3514:
3512:
3511:
3506:
3493:
3491:
3490:
3485:
3467:
3465:
3464:
3459:
3457:
3456:
3451:
3438:hyperbolic plane
3435:
3433:
3432:
3427:
3407:
3405:
3404:
3399:
3378:
3376:
3375:
3370:
3358:
3356:
3355:
3350:
3338:
3336:
3335:
3330:
3318:
3316:
3315:
3310:
3292:
3290:
3289:
3284:
3270:on its boundary
3269:
3267:
3266:
3261:
3238:
3236:
3235:
3230:
3218:
3216:
3215:
3210:
3198:
3196:
3195:
3190:
3164:
3162:
3161:
3156:
3145:
3144:
3134:
3112:
3110:
3109:
3104:
3074:
3072:
3071:
3066:
3055:
3054:
3044:
3022:
3020:
3019:
3014:
2990:
2988:
2987:
2982:
2974:
2973:
2945:
2943:
2942:
2937:
2919:
2917:
2916:
2911:
2899:
2897:
2896:
2891:
2871:
2869:
2868:
2863:
2842:
2840:
2839:
2834:
2818:
2816:
2815:
2810:
2789:
2787:
2786:
2781:
2765:
2763:
2762:
2757:
2741:
2739:
2738:
2733:
2721:
2719:
2718:
2713:
2693:
2691:
2690:
2685:
2680:
2679:
2667:
2666:
2641:
2639:
2638:
2633:
2631:
2630:
2615:
2614:
2604:
2579:
2577:
2576:
2571:
2566:
2565:
2534:
2532:
2531:
2526:
2524:
2523:
2508:
2507:
2497:
2475:
2473:
2472:
2467:
2462:
2461:
2430:
2428:
2427:
2422:
2417:
2416:
2404:
2403:
2376:
2375:
2359:
2357:
2356:
2351:
2337:
2336:
2320:
2318:
2317:
2312:
2294:
2292:
2291:
2286:
2275:
2274:
2264:
2240:
2239:
2229:
2207:
2205:
2204:
2199:
2181:
2179:
2178:
2173:
2155:
2153:
2152:
2147:
2126:
2124:
2123:
2118:
2104:
2103:
2081:
2079:
2078:
2073:
2055:
2053:
2052:
2047:
2035:
2033:
2032:
2027:
2013:
2012:
1990:
1988:
1987:
1982:
1965:) and the group
1964:
1962:
1961:
1956:
1938:
1936:
1935:
1930:
1925:
1924:
1909:
1908:
1884:
1883:
1867:
1865:
1864:
1859:
1857:
1856:
1840:
1838:
1837:
1832:
1820:
1818:
1817:
1812:
1787:
1785:
1784:
1779:
1767:
1765:
1764:
1759:
1745:
1744:
1728:
1726:
1725:
1720:
1708:
1706:
1705:
1700:
1689:(3) The element
1681:
1679:
1678:
1673:
1661:
1659:
1658:
1653:
1639:
1638:
1622:
1620:
1619:
1614:
1602:
1600:
1599:
1594:
1583:(2) The element
1575:
1573:
1572:
1567:
1555:
1553:
1552:
1547:
1535:
1533:
1532:
1527:
1516:(1) The element
1512:
1510:
1509:
1504:
1486:
1484:
1483:
1478:
1466:
1464:
1463:
1458:
1437:
1435:
1434:
1429:
1414:
1412:
1411:
1406:
1394:
1392:
1391:
1386:
1374:
1372:
1371:
1366:
1350:
1348:
1347:
1342:
1327:
1325:
1324:
1319:
1304:
1302:
1301:
1296:
1282:The action of a
1278:
1276:
1275:
1270:
1255:
1253:
1252:
1247:
1233:The action of a
1225:
1223:
1222:
1217:
1215:
1214:
1209:
1197:
1196:
1191:
1178:
1176:
1175:
1170:
1156:The action of a
1142:
1140:
1139:
1134:
1113:
1111:
1110:
1105:
1093:
1091:
1090:
1085:
1073:
1071:
1070:
1065:
1047:
1045:
1044:
1039:
1027:
1025:
1024:
1019:
998:
996:
995:
990:
952:
951:
893:
891:
890:
885:
868:
867:
836:
834:
833:
828:
816:
814:
813:
808:
796:
794:
793:
788:
768:
766:
765:
760:
758:
757:
756:
755:
734:
732:
731:
726:
702:
700:
699:
694:
677:
676:
675:
674:
654:
652:
651:
646:
641:
640:
618:
616:
615:
610:
602:
601:
585:
583:
582:
577:
547:
545:
544:
539:
527:
525:
524:
519:
507:
505:
504:
499:
487:
485:
484:
479:
467:
465:
464:
459:
435:
433:
432:
427:
425:
424:
407:
406:
399:
398:
397:
396:
375:
373:
372:
367:
343:
341:
340:
335:
309:
308:
307:
306:
285:
283:
282:
277:
269:
268:
244:
242:
241:
236:
216:
214:
213:
208:
196:
194:
193:
188:
147:
145:
144:
139:
137:
136:
131:
116:
114:
113:
108:
106:
105:
100:
79:
77:
76:
71:
59:metrizable space
46:
44:
43:
38:
4537:
4536:
4532:
4531:
4530:
4528:
4527:
4526:
4497:
4496:
4495:
4494:
4452:
4451:
4447:
4423:
4422:
4418:
4394:
4393:
4389:
4379:
4377:
4369:
4364:
4363:
4359:
4349:
4347:
4335:
4330:
4329:
4325:
4293:
4292:
4288:
4273:10.2307/2946597
4249:
4248:
4244:
4217:
4216:
4212:
4197:
4173:Gromov, Mikhail
4171:
4170:
4166:
4136:
4135:
4128:
4104:
4103:
4099:
4074:Bowditch, B. H.
4072:
4071:
4067:
4060:
4038:Bowditch, B. H.
4036:
4035:
4022:
3988:
3987:
3983:
3978:
3954:
3926:
3921:
3920:
3901:
3900:
3873:
3868:
3867:
3848:
3847:
3846:, says that if
3824:
3819:
3818:
3815:James W. Cannon
3807:
3783:
3782:
3759:
3758:
3735:
3730:
3729:
3710:
3709:
3705:by isometries.
3683:
3678:
3677:
3658:
3657:
3634:
3629:
3628:
3609:
3608:
3568:
3550:
3545:
3544:
3525:
3524:
3501:
3496:
3495:
3476:
3475:
3446:
3441:
3440:
3418:
3417:
3414:
3381:
3380:
3361:
3360:
3341:
3340:
3321:
3320:
3301:
3300:
3272:
3271:
3252:
3251:
3245:
3221:
3220:
3201:
3200:
3181:
3180:
3136:
3115:
3114:
3077:
3076:
3046:
3025:
3024:
2993:
2992:
2965:
2960:
2959:
2922:
2921:
2902:
2901:
2882:
2881:
2878:
2845:
2844:
2825:
2824:
2792:
2791:
2772:
2771:
2748:
2747:
2746:(in which case
2724:
2723:
2704:
2703:
2700:
2671:
2658:
2644:
2643:
2622:
2606:
2582:
2581:
2557:
2537:
2536:
2515:
2499:
2478:
2477:
2453:
2433:
2432:
2408:
2395:
2367:
2362:
2361:
2328:
2323:
2322:
2297:
2296:
2266:
2231:
2210:
2209:
2184:
2183:
2182:then for every
2158:
2157:
2132:
2131:
2095:
2084:
2083:
2058:
2057:
2038:
2037:
2004:
1993:
1992:
1967:
1966:
1941:
1940:
1916:
1900:
1875:
1870:
1869:
1848:
1843:
1842:
1823:
1822:
1797:
1796:
1770:
1769:
1736:
1731:
1730:
1711:
1710:
1691:
1690:
1664:
1663:
1630:
1625:
1624:
1605:
1604:
1585:
1584:
1558:
1557:
1556:; in this case
1538:
1537:
1518:
1517:
1489:
1488:
1469:
1468:
1449:
1448:
1445:
1417:
1416:
1397:
1396:
1377:
1376:
1357:
1356:
1333:
1332:
1307:
1306:
1287:
1286:
1258:
1257:
1238:
1237:
1204:
1186:
1181:
1180:
1161:
1160:
1153:
1116:
1115:
1096:
1095:
1076:
1075:
1056:
1055:
1030:
1029:
1001:
1000:
943:
896:
895:
859:
839:
838:
819:
818:
799:
798:
779:
778:
775:
747:
742:
737:
736:
705:
704:
666:
661:
656:
655:
632:
621:
620:
593:
588:
587:
550:
549:
530:
529:
510:
509:
490:
489:
470:
469:
438:
437:
400:
388:
383:
378:
377:
346:
345:
298:
293:
288:
287:
260:
255:
254:
227:
226:
199:
198:
179:
178:
175:
126:
121:
120:
95:
90:
89:
62:
61:
29:
28:
12:
11:
5:
4535:
4533:
4525:
4524:
4519:
4514:
4509:
4499:
4498:
4493:
4492:
4455:Kapovich, Ilya
4453:Jeon, Woojin;
4445:
4434:(1): 137–169.
4416:
4405:(566): 41–89.
4387:
4357:
4323:
4304:(3): 441–456.
4286:
4267:(3): 447–510.
4242:
4210:
4195:
4164:
4151:(3): 643–667.
4126:
4115:(501): 71–98.
4097:
4065:
4058:
4020:
4014:2027.42/135296
3999:(2): 331–358.
3980:
3979:
3977:
3974:
3973:
3972:
3965:
3962:
3953:
3950:
3935:
3930:
3908:
3882:
3877:
3855:
3833:
3828:
3806:
3803:
3790:
3766:
3744:
3739:
3717:
3692:
3687:
3665:
3643:
3638:
3616:
3586:
3583:
3580:
3575:
3571:
3567:
3564:
3559:
3554:
3532:
3510:
3505:
3483:
3455:
3450:
3425:
3413:
3410:
3397:
3394:
3391:
3388:
3368:
3348:
3328:
3308:
3282:
3279:
3259:
3244:
3241:
3228:
3208:
3188:
3154:
3151:
3148:
3143:
3139:
3133:
3130:
3127:
3123:
3102:
3099:
3096:
3093:
3090:
3087:
3084:
3075:and for every
3064:
3061:
3058:
3053:
3049:
3043:
3040:
3037:
3033:
3012:
3009:
3006:
3003:
3000:
2980:
2977:
2972:
2968:
2935:
2932:
2929:
2909:
2889:
2877:
2874:
2861:
2858:
2855:
2852:
2832:
2808:
2805:
2802:
2799:
2779:
2755:
2731:
2711:
2699:
2696:
2683:
2678:
2674:
2670:
2665:
2661:
2657:
2654:
2651:
2629:
2625:
2621:
2618:
2613:
2609:
2603:
2600:
2597:
2594:
2590:
2569:
2564:
2560:
2556:
2553:
2550:
2547:
2544:
2535:and for every
2522:
2518:
2514:
2511:
2506:
2502:
2496:
2493:
2490:
2486:
2465:
2460:
2456:
2452:
2449:
2446:
2443:
2440:
2420:
2415:
2411:
2407:
2402:
2398:
2394:
2391:
2388:
2385:
2382:
2379:
2374:
2370:
2349:
2346:
2343:
2340:
2335:
2331:
2310:
2307:
2304:
2284:
2281:
2278:
2273:
2269:
2263:
2260:
2257:
2254:
2250:
2246:
2243:
2238:
2234:
2228:
2225:
2222:
2218:
2197:
2194:
2191:
2171:
2168:
2165:
2145:
2142:
2139:
2116:
2113:
2110:
2107:
2102:
2098:
2094:
2091:
2071:
2068:
2065:
2045:
2025:
2022:
2019:
2016:
2011:
2007:
2003:
2000:
1980:
1977:
1974:
1954:
1951:
1948:
1928:
1923:
1919:
1915:
1912:
1907:
1903:
1899:
1896:
1893:
1890:
1887:
1882:
1878:
1855:
1851:
1830:
1810:
1807:
1804:
1777:
1757:
1754:
1751:
1748:
1743:
1739:
1718:
1698:
1671:
1651:
1648:
1645:
1642:
1637:
1633:
1612:
1592:
1565:
1545:
1525:
1502:
1499:
1496:
1476:
1456:
1444:
1441:
1440:
1439:
1427:
1424:
1404:
1384:
1364:
1340:
1329:
1317:
1314:
1294:
1280:
1268:
1265:
1245:
1231:
1213:
1208:
1203:
1200:
1195:
1190:
1168:
1158:Kleinian group
1152:
1149:
1132:
1129:
1126:
1123:
1103:
1083:
1063:
1037:
1017:
1014:
1011:
1008:
988:
985:
982:
979:
976:
973:
970:
967:
964:
961:
958:
955:
950:
946:
942:
939:
936:
933:
930:
927:
924:
921:
918:
915:
912:
909:
906:
903:
883:
880:
877:
874:
871:
866:
862:
858:
855:
852:
849:
846:
826:
806:
786:
774:
771:
754:
750:
745:
724:
721:
718:
715:
712:
692:
689:
686:
683:
680:
673:
669:
664:
644:
639:
635:
631:
628:
608:
605:
600:
596:
575:
572:
569:
566:
563:
560:
557:
537:
517:
497:
477:
457:
454:
451:
448:
445:
423:
420:
417:
414:
411:
405:
395:
391:
386:
365:
362:
359:
356:
353:
333:
330:
327:
324:
321:
318:
315:
312:
305:
301:
296:
275:
272:
267:
263:
234:
206:
186:
174:
171:
135:
130:
104:
99:
82:Kleinian group
69:
52:homeomorphisms
36:
13:
10:
9:
6:
4:
3:
2:
4534:
4523:
4520:
4518:
4515:
4513:
4510:
4508:
4505:
4504:
4502:
4487:
4482:
4477:
4472:
4468:
4464:
4460:
4456:
4449:
4446:
4441:
4437:
4433:
4429:
4428:
4420:
4417:
4412:
4408:
4404:
4400:
4399:
4391:
4388:
4380:September 12,
4375:
4368:
4361:
4358:
4350:September 12,
4345:
4341:
4334:
4327:
4324:
4319:
4315:
4311:
4307:
4303:
4299:
4298:
4290:
4287:
4282:
4278:
4274:
4270:
4266:
4262:
4261:
4256:
4252:
4246:
4243:
4238:
4234:
4230:
4226:
4225:
4220:
4214:
4211:
4206:
4202:
4198:
4196:0-387-96618-8
4192:
4187:
4182:
4178:
4174:
4168:
4165:
4159:
4154:
4150:
4146:
4145:
4140:
4133:
4131:
4127:
4122:
4118:
4114:
4110:
4109:
4101:
4098:
4093:
4089:
4085:
4081:
4080:
4075:
4069:
4066:
4061:
4059:9783110806861
4055:
4051:
4047:
4043:
4039:
4033:
4031:
4029:
4027:
4025:
4021:
4015:
4010:
4006:
4002:
3998:
3994:
3993:
3985:
3982:
3975:
3970:
3966:
3963:
3960:
3956:
3955:
3951:
3949:
3933:
3906:
3898:
3880:
3853:
3831:
3816:
3812:
3804:
3802:
3788:
3780:
3764:
3742:
3715:
3706:
3690:
3663:
3641:
3614:
3606:
3602:
3600:
3584:
3578:
3573:
3569:
3562:
3557:
3530:
3508:
3481:
3473:
3472:
3453:
3439:
3423:
3411:
3409:
3395:
3386:
3366:
3346:
3326:
3306:
3298:
3294:
3280:
3257:
3250:
3242:
3240:
3226:
3206:
3177:
3175:
3171:
3166:
3152:
3149:
3146:
3141:
3137:
3125:
3097:
3088:
3085:
3082:
3062:
3059:
3056:
3051:
3047:
3035:
3010:
3007:
3004:
3001:
2998:
2975:
2970:
2966:
2957:
2953:
2949:
2933:
2930:
2927:
2907:
2875:
2873:
2856:
2822:
2803:
2769:
2745:
2729:
2697:
2695:
2676:
2672:
2668:
2663:
2659:
2649:
2627:
2623:
2619:
2616:
2611:
2607:
2598:
2592:
2562:
2558:
2548:
2545:
2542:
2520:
2516:
2512:
2509:
2504:
2500:
2488:
2458:
2454:
2444:
2441:
2438:
2413:
2409:
2405:
2400:
2396:
2389:
2383:
2377:
2372:
2368:
2344:
2338:
2333:
2329:
2305:
2302:
2282:
2279:
2276:
2271:
2267:
2258:
2252:
2244:
2241:
2236:
2232:
2220:
2195:
2192:
2189:
2169:
2166:
2163:
2140:
2137:
2128:
2111:
2105:
2100:
2096:
2089:
2066:
2043:
2020:
2014:
2009:
2005:
1998:
1975:
1952:
1949:
1946:
1921:
1917:
1910:
1905:
1901:
1897:
1891:
1885:
1880:
1876:
1853:
1849:
1828:
1821:the elements
1808:
1805:
1802:
1793:
1791:
1775:
1752:
1746:
1741:
1737:
1696:
1687:
1685:
1669:
1646:
1640:
1635:
1631:
1590:
1581:
1579:
1563:
1523:
1514:
1497:
1494:
1474:
1442:
1425:
1382:
1354:
1338:
1330:
1315:
1292:
1285:
1281:
1266:
1243:
1236:
1232:
1229:
1211:
1198:
1193:
1159:
1155:
1154:
1150:
1148:
1146:
1127:
1081:
1052:
1049:
1035:
1012:
983:
980:
974:
971:
968:
965:
962:
953:
948:
944:
940:
934:
931:
928:
925:
922:
913:
907:
878:
864:
860:
856:
850:
824:
804:
772:
770:
752:
748:
743:
722:
719:
716:
713:
710:
690:
687:
681:
671:
667:
662:
642:
637:
633:
629:
626:
606:
603:
598:
594:
570:
561:
558:
555:
535:
515:
495:
475:
452:
443:
418:
409:
393:
389:
384:
363:
360:
357:
354:
351:
331:
328:
325:
322:
319:
316:
313:
310:
303:
299:
294:
270:
265:
261:
252:
248:
224:
220:
204:
172:
170:
168:
164:
160:
156:
152:
148:
133:
102:
87:
83:
67:
60:
57:
53:
49:
27:
23:
19:
4507:Group theory
4469:(4): 58–80.
4466:
4462:
4448:
4431:
4425:
4419:
4402:
4396:
4390:
4378:. Retrieved
4373:
4360:
4348:. Retrieved
4346:(2): 333–348
4343:
4342:. Series A.
4339:
4326:
4301:
4295:
4289:
4264:
4258:
4251:Gabai, Davis
4245:
4228:
4222:
4219:Tukia, Pekka
4213:
4176:
4167:
4148:
4142:
4112:
4106:
4100:
4083:
4077:
4068:
4041:
3996:
3990:
3984:
3808:
3757:, the group
3707:
3604:
3603:
3470:
3415:
3296:
3295:
3246:
3178:
3167:
2955:
2951:
2947:
2946:is called a
2879:
2767:
2766:is called a
2743:
2701:
2129:
1794:
1789:
1688:
1683:
1582:
1577:
1515:
1446:
1053:
1050:
817:. For a set
776:
250:
246:
245:is called a
222:
218:
176:
21:
17:
15:
4231:: 318–346.
344:and points
4501:Categories
3976:References
3468:is called
3176:, states:
3023:such that
2821:co-compact
2742:is called
1790:loxodromic
1788:is called
1682:is called
1576:is called
999:. The set
225:(and then
4476:1401.2638
3779:virtually
3582:∂
3579:≈
3566:∂
3471:geometric
3393:∂
3390:→
3278:∂
3187:Γ
3138:γ
3132:∞
3129:→
3092:∖
3086:∈
3048:γ
3042:∞
3039:→
3008:∈
2979:Γ
2976:∈
2967:γ
2931:∈
2888:Γ
2851:Θ
2831:Γ
2798:Θ
2778:Γ
2754:Γ
2710:Γ
2664:−
2653:∖
2628:−
2608:γ
2602:∞
2599:−
2596:→
2552:∖
2546:∈
2501:γ
2495:∞
2492:→
2459:−
2448:∖
2442:∈
2401:−
2384:γ
2378:
2345:γ
2339:
2309:Γ
2306:∈
2303:γ
2268:γ
2262:∞
2259:−
2256:→
2233:γ
2227:∞
2224:→
2193:∈
2167:∈
2144:Γ
2141:∈
2138:γ
2112:γ
2106:
2093:∖
2070:⟩
2067:γ
2064:⟨
2044:γ
2021:γ
2015:
2002:∖
1979:⟩
1976:γ
1973:⟨
1950:≠
1918:γ
1911:
1892:γ
1886:
1850:γ
1829:γ
1806:≠
1776:γ
1753:γ
1747:
1717:Γ
1697:γ
1684:parabolic
1670:γ
1647:γ
1641:
1611:Γ
1591:γ
1564:γ
1544:Γ
1524:γ
1501:Γ
1498:∈
1495:γ
1455:Γ
1423:∂
1403:Γ
1363:Γ
1313:∂
1264:∂
1202:∂
1167:Γ
1122:Θ
1102:Γ
1062:Γ
1007:Θ
981:≤
957:#
954:∣
941:∈
902:Δ
873:Δ
870:∖
845:Θ
785:Γ
744:γ
720:∈
688:⊆
663:γ
630:≥
604:≥
565:∖
559:⊂
447:∖
413:∖
385:γ
361:∈
332:…
295:γ
274:Γ
271:∈
262:γ
233:Γ
185:Γ
35:Γ
4253:(1992).
3605:Theorem.
3297:Theorem.
3174:Bowditch
3113:one has
2580:one has
2476:one has
2208:one has
1578:elliptic
1151:Examples
894:, where
4306:Bibcode
4281:2946597
4205:0919829
4086:(662).
3436:on the
2823:. Thus
2744:uniform
1939:(where
837:denote
151:Gehring
117:of the
56:compact
4279:
4203:
4193:
4056:
2954:or a
165:, and
155:Martin
48:acting
4471:arXiv
4370:(PDF)
4336:(PDF)
4277:JSTOR
3599:Gabai
3170:Tukia
249:or a
221:or a
54:on a
26:group
24:is a
20:or a
4403:2004
4382:2022
4352:2022
4191:ISBN
4113:1998
4054:ISBN
3494:on
3299:Let
2880:Let
2130:If
1841:and
1447:Let
1331:Let
1054:Let
177:Let
153:and
4481:doi
4436:doi
4407:doi
4314:doi
4302:118
4269:doi
4265:136
4233:doi
4181:doi
4153:doi
4117:doi
4088:doi
4084:139
4046:doi
4009:hdl
4001:doi
3919:on
3899:of
3777:is
3676:on
3607:If
3543:on
3122:lim
3032:lim
2819:is
2790:on
2589:lim
2485:lim
2369:Fix
2330:Fix
2295:If
2249:lim
2217:lim
2097:Fix
2006:Fix
1902:Fix
1877:Fix
1738:Fix
1632:Fix
1415:on
1226:by
1179:on
1143:is
1114:on
528:in
508:of
468:to
84:by
50:by
4503::
4479:.
4467:20
4465:.
4461:.
4432:19
4430:.
4401:.
4372:.
4344:20
4338:.
4312:.
4300:.
4275:.
4257:.
4229:46
4227:.
4201:MR
4199:.
4189:.
4149:11
4147:.
4141:.
4129:^
4111:.
4082:.
4052:.
4023:^
4007:.
3997:55
3995:.
3408:.
3165:.
2694:.
2127:.
1792:.
1686:.
1580:.
1147:.
1048:.
857::=
169:.
161:,
4489:.
4483::
4473::
4442:.
4438::
4413:.
4409::
4384:.
4354:.
4320:.
4316::
4308::
4283:.
4271::
4239:.
4235::
4207:.
4183::
4161:.
4155::
4123:.
4119::
4094:.
4090::
4062:.
4048::
4017:.
4011::
4003::
3934:3
3929:H
3907:G
3881:2
3876:S
3854:G
3832:2
3827:S
3789:G
3765:G
3743:2
3738:H
3716:G
3691:2
3686:H
3664:G
3642:1
3637:S
3615:G
3585:G
3574:2
3570:H
3563:=
3558:1
3553:S
3531:G
3509:2
3504:H
3482:G
3454:2
3449:H
3424:G
3396:G
3387:M
3367:G
3347:G
3327:M
3307:G
3281:G
3258:G
3227:M
3207:M
3153:b
3150:=
3147:y
3142:n
3126:n
3101:}
3098:x
3095:{
3089:M
3083:y
3063:a
3060:=
3057:x
3052:n
3036:n
3011:M
3005:b
3002:,
2999:a
2971:n
2934:M
2928:x
2908:M
2860:)
2857:M
2854:(
2807:)
2804:M
2801:(
2730:M
2682:}
2677:+
2673:a
2669:,
2660:a
2656:{
2650:M
2624:a
2620:=
2617:x
2612:n
2593:n
2568:}
2563:+
2559:a
2555:{
2549:M
2543:x
2521:+
2517:a
2513:=
2510:x
2505:n
2489:n
2464:}
2455:a
2451:{
2445:M
2439:x
2419:}
2414:+
2410:a
2406:,
2397:a
2393:{
2390:=
2387:)
2381:(
2373:M
2348:)
2342:(
2334:M
2283:a
2280:=
2277:x
2272:n
2253:n
2245:=
2242:x
2237:n
2221:n
2196:M
2190:x
2170:M
2164:a
2115:)
2109:(
2101:M
2090:M
2024:)
2018:(
2010:M
1999:M
1953:0
1947:p
1927:)
1922:p
1914:(
1906:M
1898:=
1895:)
1889:(
1881:M
1854:p
1809:0
1803:p
1756:)
1750:(
1742:M
1650:)
1644:(
1636:M
1475:M
1426:X
1383:X
1339:X
1316:G
1293:G
1267:G
1244:G
1212:3
1207:H
1199:=
1194:2
1189:S
1131:)
1128:M
1125:(
1082:M
1036:M
1016:)
1013:M
1010:(
987:}
984:2
978:}
975:c
972:,
969:b
966:,
963:a
960:{
949:3
945:M
938:)
935:c
932:,
929:b
926:,
923:a
920:(
917:{
914:=
911:)
908:M
905:(
882:)
879:M
876:(
865:3
861:M
854:)
851:M
848:(
825:M
805:M
753:k
749:n
723:M
717:b
714:,
711:a
691:U
685:)
682:K
679:(
672:k
668:n
643:,
638:0
634:k
627:k
607:1
599:0
595:k
574:}
571:a
568:{
562:M
556:K
536:M
516:b
496:U
476:b
456:}
453:a
450:{
444:M
422:}
419:a
416:{
410:M
404:|
394:k
390:n
364:M
358:b
355:,
352:a
329:,
326:2
323:,
320:1
317:=
314:k
311:,
304:k
300:n
266:n
205:M
134:3
129:H
103:2
98:S
68:M
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