329:
1674:
550:
495:
915:
1532:
1030:
1573:
992:
950:
149:
2024:
1285:
1176:
705:
666:
222:
1375:
1727:
1075:
440:
181:
1489:
1404:
1337:
1311:
807:
358:
111:
857:
1679:
Many groups admitting nontrivial actions on simplicial trees (that is, admitting nontrivial splittings as fundamental groups of graphs of groups in the sense of
831:
2232:
1881:
1790:
1798:
230:
1581:
2116:
1683:) are acylindrically hyperbolic. For example, all one-relator groups on at least three generators are acylindrically hyperbolic.
2269:
1930:
500:
445:
1122:
over acylindrically hyperbolic groups, allowing one to produce many quotients of such groups with prescribed properties.
1440:
is a finitely generated group acting isometrically properly discontinuously and cocompactly on a geodetically complete
2274:
1230:
41:
1576:
29:
1977:
1119:
862:
2075:
403:
388:
77:
33:
1820:
1219:
Finite groups, virtually nilpotent groups and virtually solvable groups are not acylindrically hyperbolic.
715:
17:
1680:
2264:
2174:
1497:
1223:
1033:
997:
1540:
1492:
1436:
having at least one rank-1 element, is acylindrically hyperbolic. Caprace and Sageev proved that if
955:
927:
595:
admits a non-elementary acylindrical isometric action on some geodesic hyperbolic metric space
1237:
116:
45:
25:
1242:
1155:
675:
636:
186:
2201:
2183:
2151:
2125:
2084:
2051:
2033:
2004:
1986:
1957:
1939:
1910:
1890:
1857:
1829:
1762:
1736:
1342:
1137:
765:
1038:
2279:
1794:
1186:
413:
154:
1787:
Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces
1468:
1383:
1316:
1290:
786:
337:
90:
2241:
2224:
2193:
2135:
2094:
2043:
1996:
1949:
1900:
1839:
1746:
836:
37:
2147:
1853:
1758:
2143:
1849:
1754:
1126:
780:
1462:, which is not cyclic and which is directly indecomposable, is acylindrically hyperbolic.
2172:
Minasyan, A.; Osin, D. (2015). "Acylindrical hyperbolicity of groups acting on trees".
816:
769:
1125:
Every finitely generated acylindrically hyperbolic group has cut points in all of its
2258:
2155:
2055:
2008:
1961:
571:
is not a finite extension of a cyclic group generated by loxodromic isometry of
377:
2205:
1914:
1861:
1766:
1441:
1429:
1198:
631:
2197:
2139:
2099:
2070:
2000:
1421:
1108:
779:
The class of acylindrically hyperbolic groups is closed under taking infinite
2114:
Caprace, P.E.; Sageev, M. (2011). "Rank rigidity for CAT(0) cube complexes".
2047:
1928:
Hull, M. (2016). "Small cancellation in acylindrically hyperbolic groups".
1975:
Sisto, A. (2016). "Quasi-convexity of hyperbolically embedded subgroups".
783:, and, more generally, under taking 's-normal' subgroups. Here a subgroup
376:. The notion of acylindricity provides a suitable substitute for being a
1844:
1815:
742:
1750:
1905:
1876:
1407:
49:
1953:
1185:
has exponential conjugacy growth, meaning that the number of distinct
2246:
1895:
1834:
563:
is non-elementary if and only if this action has unbounded orbits in
555:
It is known (Theorem 1.1 in ) that an acylindrical action of a group
1451:
splits as a direct product of two unbounded convex subcomplexes, or
1339:
punctures is acylindrically hyperbolic, except for the cases where
380:
in the more general context where non-proper actions are allowed.
324:{\displaystyle \#\{g\in G\mid d(x,gx)\leq R,d(y,gy)\leq R\}\leq N.}
2188:
2130:
2089:
2038:
1991:
1944:
1741:
1669:{\displaystyle BS(m,n)=\langle a,t\mid t^{-1}a^{m}t=a^{n}\rangle }
1377:(in those exceptional cases the mapping class group is finite).
768:, that is, every countable group embeds as a subgroup in some
44:
and includes a significantly wider class of examples, such as
1877:"Bounded cohomology and isometry groups of hyperbolic spaces"
2022:
Sisto, A. (2018). "Contracting elements and random walks".
1816:"Bounded cohomology of subgroups of mapping class groups"
1181:
Every finitely generated acylindrically hyperbolic group
1132:
For a finitely generated acylindrically hyperbolic group
734:
with the WPD ('Weakly
Properly Discontinuous') property.
1103:
has infinite conjugacy classes of nontrivial elements,
1084:
admits a unique maximal normal finite subgroup denoted
1725:
Osin, D. (2016). "Acylindrically hyperbolic groups".
1686:
Most 3-manifold groups are acylindrically hyperbolic.
1584:
1543:
1500:
1471:
1386:
1345:
1319:
1293:
1245:
1158:
1041:
1000:
958:
930:
865:
839:
819:
789:
741:
contains a proper infinite 'hyperbolically embedded'
678:
668:
is hyperbolic, and the natural translation action of
639:
503:
448:
416:
340:
233:
189:
157:
119:
93:
76:
be a group with an isometric action on some geodesic
1428:, that admits a proper isometric action on a proper
28:
admitting a non-elementary 'acylindrical' isometric
1676:
is not acylindrically hyperbolic. (Example 7.4 in )
1534:
is not acylindrically hyperbolic (Example 7.5 in ).
545:{\displaystyle \{h^{+},h^{-}\}\subseteq \partial X}
490:{\displaystyle \{g^{+},g^{-}\}\subseteq \partial X}
1668:
1567:
1526:
1483:
1398:
1369:
1331:
1305:
1279:
1170:
1069:
1024:
986:
944:
909:
851:
825:
801:
699:
660:
622:There exists a (possibly infinite) generating set
544:
489:
434:
352:
323:
216:
175:
143:
105:
1728:Transactions of the American Mathematical Society
1793:. Vol. 245. American Mathematical Society.
2071:"Conjugacy growth of finitely generated groups"
2025:Journal fĂĽr die reine und angewandte Mathematik
1148:produces a 'generalized loxodromic element' in
607:It is known (Theorem 1.2 in ) that for a group
1785:Dahmani, F.; Guirardel, V.; Osin, D. (2017).
8:
2233:Notices of the American Mathematical Society
1882:Journal of the European Mathematical Society
1791:Memoirs of the American Mathematical Society
1663:
1609:
530:
504:
475:
449:
383:An acylindrical isometric action of a group
309:
237:
1095:is an acylindrically hyperbolic group with
334:If the above property holds for a specific
36:. This notion generalizes the notions of a
924:is an acylindrically hyperbolic group and
718:, and there exists an isometric action of
2245:
2225:"WHAT IS...an Acylindrical Group Action?"
2187:
2129:
2098:
2088:
2037:
1990:
1943:
1904:
1894:
1843:
1833:
1740:
1657:
1641:
1628:
1583:
1542:
1517:
1516:
1499:
1470:
1385:
1344:
1318:
1292:
1287:of a connected oriented surface of genus
1262:
1244:
1157:
1046:
1040:
999:
969:
957:
938:
937:
929:
896:
881:
866:
864:
838:
818:
788:
677:
638:
611:the following conditions are equivalent:
524:
511:
502:
469:
456:
447:
415:
339:
232:
188:
156:
118:
92:
910:{\displaystyle |H\cap g^{-1}Hg|=\infty }
707:is a non-elementary acylindrical action.
1720:
1696:
1107:is not inner amenable, and the reduced
1718:
1716:
1714:
1712:
1710:
1708:
1706:
1704:
1702:
1700:
1080:Every acylindrically hyperbolic group
760:Every acylindrically hyperbolic group
722:on a geodesic hyperbolic metric space
559:on a geodesic hyperbolic metric space
2167:
2165:
1780:
1778:
1776:
1152:converges to 1 exponentially fast as
7:
1814:Bestvina, M.; Fujiwara, K. (2002).
1424:, every non virtually cyclic group
1222:Every non-elementary subgroup of a
410:, that is, two loxodromic elements
1527:{\displaystyle SL(n,\mathbb {Z} )}
1165:
1016:
904:
726:such that at least one element of
679:
640:
536:
481:
234:
14:
2117:Geometric and Functional Analysis
442:such that their fixed point sets
1025:{\displaystyle p\in [1,\infty )}
1568:{\displaystyle m\neq 0,n\neq 0}
1458:Every right-angled Artin group
1193:coming from the ball of radius
579:Acylindrically hyperbolic group
22:acylindrically hyperbolic group
16:In the mathematical subject of
1931:Groups, Geometry, and Dynamics
1603:
1591:
1521:
1507:
1274:
1255:
1162:
1064:
1052:
1019:
1007:
987:{\displaystyle V=\ell ^{p}(G)}
981:
975:
945:{\displaystyle V=\mathbb {R} }
897:
867:
694:
682:
655:
643:
300:
285:
270:
255:
205:
193:
1:
1417:is acylindrically hyperbolic.
1233:is acylindrically hyperbolic.
1226:is acylindrically hyperbolic.
619:is acylindrically hyperbolic.
144:{\displaystyle N>0,L>0}
1280:{\displaystyle MCG(S_{g,p})}
1171:{\displaystyle n\to \infty }
1115:is simple with unique trace.
700:{\displaystyle \Gamma (G,S)}
661:{\displaystyle \Gamma (G,S)}
603:Equivalent characterizations
217:{\displaystyle d(x,y)\geq L}
2069:Hull, M.; Osin, D. (2013).
1455:contains a rank-1 element.
1370:{\displaystyle g=0,p\leq 3}
1231:relatively hyperbolic group
1136:, the probability that the
42:relatively hyperbolic group
2296:
1070:{\displaystyle H_{b}(G,V)}
2198:10.1007/s00208-014-1138-z
2140:10.1007/s00039-011-0126-7
2100:10.1016/j.aim.2012.12.007
2001:10.1007/s00209-016-1615-z
1978:Mathematische Zeitschrift
1214:Examples and non-examples
1120:small cancellation theory
589:acylindrically hyperbolic
2223:Koberda, Thomas (2018).
2048:10.1515/crelle-2015-0093
1077:is infinite-dimensional.
435:{\displaystyle g,h\in G}
176:{\displaystyle x,y\in X}
83:. This action is called
2076:Advances in Mathematics
1875:Hamenstädt, U. (2008).
1821:Geometry & Topology
1484:{\displaystyle n\geq 3}
1399:{\displaystyle n\geq 2}
1332:{\displaystyle p\geq 0}
1306:{\displaystyle g\geq 0}
1205:grows exponentially in
802:{\displaystyle H\leq G}
402:admits two independent
389:hyperbolic metric space
353:{\displaystyle R\geq 0}
106:{\displaystyle R\geq 0}
78:hyperbolic metric space
34:hyperbolic metric space
2270:Geometric group theory
1670:
1577:Baumslag–Solitar group
1569:
1528:
1485:
1442:CAT(0) cubical complex
1400:
1371:
1333:
1307:
1281:
1172:
1118:There is a version of
1071:
1026:
988:
946:
911:
853:
852:{\displaystyle g\in G}
827:
803:
701:
662:
546:
491:
436:
354:
325:
218:
177:
145:
107:
18:geometric group theory
2175:Mathematische Annalen
1671:
1570:
1529:
1486:
1401:
1372:
1334:
1308:
1282:
1229:Every non-elementary
1224:word-hyperbolic group
1173:
1072:
1027:
989:
947:
912:
854:
828:
804:
702:
663:
547:
492:
437:
355:
326:
219:
178:
146:
108:
1845:10.2140/gt.2002.6.69
1582:
1541:
1498:
1493:special linear group
1469:
1384:
1343:
1317:
1291:
1243:
1156:
1039:
998:
956:
928:
863:
837:
817:
787:
676:
637:
501:
446:
414:
338:
231:
187:
155:
151:such that for every
117:
91:
46:mapping class groups
1238:mapping class group
68:Acylindrical action
2275:Geometric topology
2182:(3–4): 1055–1105.
1666:
1565:
1524:
1481:
1396:
1367:
1329:
1303:
1277:
1168:
1138:simple random walk
1067:
1034:bounded cohomology
1022:
984:
942:
907:
849:
823:
799:
697:
658:
542:
487:
432:
350:
321:
214:
173:
141:
103:
1800:978-1-4704-2194-6
1751:10.1090/tran/6343
1681:Bass–Serre theory
1187:conjugacy classes
826:{\displaystyle G}
63:Formal definition
32:on some geodesic
2287:
2251:
2249:
2247:10.1090/noti1624
2229:
2210:
2209:
2191:
2169:
2160:
2159:
2133:
2111:
2105:
2104:
2102:
2092:
2066:
2060:
2059:
2041:
2019:
2013:
2012:
1994:
1985:(3–4): 649–658.
1972:
1966:
1965:
1947:
1938:(4): 1077–1119.
1925:
1919:
1918:
1908:
1906:10.4171/JEMS/112
1898:
1872:
1866:
1865:
1847:
1837:
1811:
1805:
1804:
1782:
1771:
1770:
1744:
1722:
1675:
1673:
1672:
1667:
1662:
1661:
1646:
1645:
1636:
1635:
1574:
1572:
1571:
1566:
1533:
1531:
1530:
1525:
1520:
1490:
1488:
1487:
1482:
1405:
1403:
1402:
1397:
1376:
1374:
1373:
1368:
1338:
1336:
1335:
1330:
1312:
1310:
1309:
1304:
1286:
1284:
1283:
1278:
1273:
1272:
1177:
1175:
1174:
1169:
1127:asymptotic cones
1076:
1074:
1073:
1068:
1051:
1050:
1031:
1029:
1028:
1023:
993:
991:
990:
985:
974:
973:
951:
949:
948:
943:
941:
916:
914:
913:
908:
900:
889:
888:
870:
858:
856:
855:
850:
832:
830:
829:
824:
808:
806:
805:
800:
781:normal subgroups
716:virtually cyclic
706:
704:
703:
698:
667:
665:
664:
659:
630:, such that the
551:
549:
548:
543:
529:
528:
516:
515:
496:
494:
493:
488:
474:
473:
461:
460:
441:
439:
438:
433:
360:, the action of
359:
357:
356:
351:
330:
328:
327:
322:
223:
221:
220:
215:
182:
180:
179:
174:
150:
148:
147:
142:
112:
110:
109:
104:
38:hyperbolic group
2295:
2294:
2290:
2289:
2288:
2286:
2285:
2284:
2255:
2254:
2227:
2222:
2219:
2217:Further reading
2214:
2213:
2171:
2170:
2163:
2113:
2112:
2108:
2068:
2067:
2063:
2032:(742): 79–114.
2021:
2020:
2016:
1974:
1973:
1969:
1954:10.4171/GGD/377
1927:
1926:
1922:
1874:
1873:
1869:
1813:
1812:
1808:
1801:
1784:
1783:
1774:
1724:
1723:
1698:
1693:
1653:
1637:
1624:
1580:
1579:
1539:
1538:
1496:
1495:
1467:
1466:
1420:By a result of
1413:
1382:
1381:
1341:
1340:
1315:
1314:
1289:
1288:
1258:
1241:
1240:
1216:
1189:of elements of
1154:
1153:
1042:
1037:
1036:
996:
995:
965:
954:
953:
926:
925:
877:
861:
860:
835:
834:
815:
814:
785:
784:
757:
752:
674:
673:
635:
634:
605:
581:
552:are disjoint.
520:
507:
499:
498:
465:
452:
444:
443:
412:
411:
336:
335:
229:
228:
185:
184:
153:
152:
115:
114:
89:
88:
70:
65:
55:
12:
11:
5:
2293:
2291:
2283:
2282:
2277:
2272:
2267:
2257:
2256:
2253:
2252:
2218:
2215:
2212:
2211:
2161:
2124:(4): 851–891.
2106:
2083:(1): 361–389.
2061:
2014:
1967:
1920:
1889:(2): 315–349.
1867:
1806:
1799:
1772:
1735:(2): 851–888.
1695:
1694:
1692:
1689:
1688:
1687:
1684:
1677:
1665:
1660:
1656:
1652:
1649:
1644:
1640:
1634:
1631:
1627:
1623:
1620:
1617:
1614:
1611:
1608:
1605:
1602:
1599:
1596:
1593:
1590:
1587:
1564:
1561:
1558:
1555:
1552:
1549:
1546:
1535:
1523:
1519:
1515:
1512:
1509:
1506:
1503:
1480:
1477:
1474:
1463:
1456:
1447:, then either
1418:
1411:
1395:
1392:
1389:
1378:
1366:
1363:
1360:
1357:
1354:
1351:
1348:
1328:
1325:
1322:
1302:
1299:
1296:
1276:
1271:
1268:
1265:
1261:
1257:
1254:
1251:
1248:
1234:
1227:
1220:
1215:
1212:
1211:
1210:
1179:
1167:
1164:
1161:
1130:
1123:
1116:
1089:
1078:
1066:
1063:
1060:
1057:
1054:
1049:
1045:
1021:
1018:
1015:
1012:
1009:
1006:
1003:
983:
980:
977:
972:
968:
964:
961:
940:
936:
933:
918:
906:
903:
899:
895:
892:
887:
884:
880:
876:
873:
869:
848:
845:
842:
822:
798:
795:
792:
777:
770:quotient group
756:
753:
751:
748:
747:
746:
735:
708:
696:
693:
690:
687:
684:
681:
657:
654:
651:
648:
645:
642:
620:
604:
601:
580:
577:
567:and the group
541:
538:
535:
532:
527:
523:
519:
514:
510:
506:
486:
483:
480:
477:
472:
468:
464:
459:
455:
451:
431:
428:
425:
422:
419:
406:isometries of
396:non-elementary
387:on a geodesic
349:
346:
343:
332:
331:
320:
317:
314:
311:
308:
305:
302:
299:
296:
293:
290:
287:
284:
281:
278:
275:
272:
269:
266:
263:
260:
257:
254:
251:
248:
245:
242:
239:
236:
213:
210:
207:
204:
201:
198:
195:
192:
172:
169:
166:
163:
160:
140:
137:
134:
131:
128:
125:
122:
102:
99:
96:
69:
66:
64:
61:
53:
13:
10:
9:
6:
4:
3:
2:
2292:
2281:
2278:
2276:
2273:
2271:
2268:
2266:
2263:
2262:
2260:
2248:
2243:
2239:
2235:
2234:
2226:
2221:
2220:
2216:
2207:
2203:
2199:
2195:
2190:
2185:
2181:
2177:
2176:
2168:
2166:
2162:
2157:
2153:
2149:
2145:
2141:
2137:
2132:
2127:
2123:
2119:
2118:
2110:
2107:
2101:
2096:
2091:
2086:
2082:
2078:
2077:
2072:
2065:
2062:
2057:
2053:
2049:
2045:
2040:
2035:
2031:
2027:
2026:
2018:
2015:
2010:
2006:
2002:
1998:
1993:
1988:
1984:
1980:
1979:
1971:
1968:
1963:
1959:
1955:
1951:
1946:
1941:
1937:
1933:
1932:
1924:
1921:
1916:
1912:
1907:
1902:
1897:
1892:
1888:
1884:
1883:
1878:
1871:
1868:
1863:
1859:
1855:
1851:
1846:
1841:
1836:
1831:
1827:
1823:
1822:
1817:
1810:
1807:
1802:
1796:
1792:
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840:
833:if for every
820:
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405:
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397:
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390:
386:
381:
379:
378:proper action
375:
371:
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297:
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87:if for every
86:
82:
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75:
67:
62:
60:
58:
56:
47:
43:
39:
35:
31:
27:
23:
19:
2265:Group theory
2240:(1): 31–34.
2237:
2231:
2179:
2173:
2121:
2115:
2109:
2080:
2074:
2064:
2029:
2023:
2017:
1982:
1976:
1970:
1935:
1929:
1923:
1896:math/0507097
1886:
1880:
1870:
1835:math/0012115
1825:
1819:
1809:
1786:
1732:
1726:
1459:
1452:
1448:
1444:
1437:
1433:
1430:CAT(0) space
1425:
1409:
1206:
1202:
1199:Cayley graph
1194:
1190:
1182:
1149:
1145:
1141:
1133:
1112:
1104:
1100:
1096:
1092:
1085:
1081:
921:
810:
773:
766:SQ-universal
761:
738:
731:
727:
723:
719:
711:
669:
632:Cayley graph
627:
623:
616:
608:
606:
596:
592:
588:
584:
582:
572:
568:
564:
560:
556:
554:
407:
399:
395:
391:
384:
382:
374:acylindrical
373:
369:
365:
361:
333:
113:there exist
85:acylindrical
84:
80:
73:
71:
51:
21:
15:
2259:Categories
1691:References
1406:the group
1144:of length
1109:C*-algebra
809:is called
755:Properties
737:The group
710:The group
615:The group
587:is called
404:hyperbolic
368:is called
2189:1310.6289
2156:119326592
2131:1005.5687
2090:1107.1826
2056:118009555
2039:1112.2666
2009:119174222
1992:1310.7753
1962:118319683
1945:1308.4345
1828:: 69–89.
1742:1304.1246
1664:⟩
1630:−
1622:∣
1610:⟨
1560:≠
1548:≠
1476:≥
1391:≥
1362:≤
1324:≥
1298:≥
1166:∞
1163:→
1032:then the
1017:∞
1005:∈
967:ℓ
905:∞
883:−
875:∩
844:∈
794:≤
680:Γ
641:Γ
537:∂
534:⊆
526:−
482:∂
479:⊆
471:−
427:∈
345:≥
313:≤
304:≤
274:≤
250:∣
244:∈
235:#
209:≥
168:∈
98:≥
40:and of a
2280:Geometry
2206:55851214
1915:16750741
1862:11350501
1767:21624534
1097:K(G)={1}
859:one has
811:s-normal
743:subgroup
730:acts on
583:A group
224:one has
2148:2827012
1854:1914565
1803:. 1156.
1759:3430352
1197:in the
750:History
714:is not
2204:
2154:
2146:
2054:
2007:
1960:
1913:
1860:
1852:
1797:
1765:
1757:
30:action
2228:(PDF)
2202:S2CID
2184:arXiv
2152:S2CID
2126:arXiv
2085:arXiv
2052:S2CID
2034:arXiv
2005:S2CID
1987:arXiv
1958:S2CID
1940:arXiv
1911:S2CID
1891:arXiv
1858:S2CID
1830:arXiv
1763:S2CID
1737:arXiv
1432:with
1313:with
1099:then
994:with
183:with
26:group
24:is a
20:, an
2030:2018
1795:ISBN
1575:the
1537:For
1491:the
1465:For
1422:Osin
1408:Out(
1380:For
1236:The
1086:K(G)
626:for
497:and
136:>
124:>
72:Let
50:Out(
48:and
2242:doi
2194:doi
2180:362
2136:doi
2095:doi
2081:235
2044:doi
1997:doi
1983:283
1950:doi
1901:doi
1840:doi
1747:doi
1733:368
1201:of
1140:on
1111:of
1091:If
952:or
920:If
813:in
772:of
764:is
672:on
591:if
398:if
394:is
364:on
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1999::
1989::
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1832::
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