92:
and, moreover, it satisfies a technical condition which implies that quasi-geodesics with common endpoints travel through approximately the same collection of cosets and enter and exit these cosets in approximately the same place.
564:
283:
169:
315:
The definition of the coned off Cayley graph can be generalized to the case of a collection of subgroups and yields the corresponding notion of relative hyperbolicity. A group
319:
which contains no collection of subgroups with respect to which it is relatively hyperbolic is said to be a non relatively hyperbolic group.
457:
for virtually torsion-free relatively hyperbolic groups when the peripheral subgroups are finitely generated nilpotent (Dahmani, Touikan)
514:
582:
The semi-direct product of a free group by an infinite cyclic group is relatively hyperbolic, relative to some canonical subgroups.
441:
More generally, in many cases (but not all, and not easily or systematically), a property satisfied by all hyperbolic groups and by
701:
637:
579:
Limit groups appearing as limits of free groups are relatively hyperbolic, relative to some free abelian subgroups.
371:
454:
475:
of finite rank or the fundamental group of a hyperbolic surface, is hyperbolic relative to the trivial subgroup.
429:
102:
244:
130:
585:
Combination theorems and small cancellation techniques allow to construct new examples from previous ones.
25:
290:
89:
589:
494:
490:
486:
482:
220:
55:
43:
624:
620:
593:
501:
40:
653:
Relatively hyperbolic groups: Intrinsic geometry, algebraic properties, and algorithmic problems
670:
479:
408:
383:
354:
37:
33:
609:
468:
29:
644:, Essays in group theory, Math. Sci. Res. Inst. Publ., 8, 75-263, Springer, New York, 1987.
666:
616:
is the number of punctures) or is not relatively hyperbolic with respect to any subgroup.
300:: for each integer L, every edge belongs to only finitely many simple cycles of length L.
227:
695:
507:
of rank 2 is weakly hyperbolic, but not hyperbolic, relative to the cyclic subgroup
73:
17:
647:
627:
group of a free group of finite rank at least 3 are not relatively hyperbolic.
472:
676:
Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity
66:
674:
651:
683:
88:, the resulting graph equipped with the usual graph metric becomes a
226:
The definition of a relatively hyperbolic group, as formulated by
85:
32:. The motivating examples of relatively hyperbolic groups are the
573:
with any hyperbolic group, is relatively hyperbolic, relative to
559:{\displaystyle {\hat {\Gamma }}(\mathbb {Z} ^{2},\mathbb {Z} )}
399:
is torsion-free relatively hyperbolic with respect to a group
332:
is relatively hyperbolic with respect to a hyperbolic group
308:
is said to be weakly relatively hyperbolic with respect to
489:. A similar result holds for any complete finite volume
517:
247:
133:
558:
277:
219:). This results in a metric space that may not be
163:
436:satisfies the Farrell-jones conjecture (Bartels).
424:is relatively hyperbolic with respect to a group
366:is relatively hyperbolic with respect to a group
349:is relatively hyperbolic with respect to a group
304:If only the first condition holds then the group
679:, arXiv:math/0512592v5 (math.GT), December 2005.
485:of finite volume is hyperbolic relative to its
116:) equipped with the path metric and a subgroup
656:, arXiv:math/0404040v1 (math.GR), April 2004.
8:
685:Dehn filling in relatively hyperbolic groups
549:
548:
539:
535:
534:
519:
518:
516:
378:has solvable word problem (Farb), and if
357:on a compact space, its Bowditch boundary
249:
248:
246:
223:(i.e. closed balls need not be compact).
135:
134:
132:
390:has solvable conjugacy problem (Bumagin)
688:, arXiv:math/0601311v4 , January 2007.
663:, Geom. Funct. Anal. 8 (1998), 810–840.
353:then it acts as a geometrically finite
24:is an important generalization of the
682:Daniel Groves and Jason Fox Manning,
278:{\displaystyle {\hat {\Gamma }}(G,H)}
164:{\displaystyle {\hat {\Gamma }}(G,H)}
7:
445:can be suspected to be satisfied by
521:
251:
236:hyperbolic relative to a subgroup
137:
14:
171:as follows: For each left coset
566:is hyperbolic, it is not fine.
553:
530:
524:
272:
260:
254:
241:if the coned off Cayley graph
158:
146:
140:
1:
592:of an orientable finite type
661:Relatively hyperbolic groups
596:is either hyperbolic (when 3
569:The free product of a group
478:The fundamental group of a
22:relatively hyperbolic group
718:
72:if, after contracting the
230:goes as follows. A group
511:: even though the graph
430:Farrell-Jones conjecture
124:, one can construct the
103:finitely generated group
191:) and for each element
702:Geometric group theory
560:
493:with pinched negative
279:
183:) to the Cayley graph
165:
126:coned off Cayley graph
26:geometric group theory
561:
340:itself is hyperbolic.
280:
207:) of length 1/2 from
166:
63:relatively hyperbolic
515:
285:has the properties:
245:
131:
50:Intuitive definition
44:hyperbolic manifolds
590:mapping class group
495:sectional curvature
491:Riemannian manifold
483:hyperbolic manifold
455:isomorphism problem
428:that satisfies the
20:, the concept of a
625:outer automorphism
621:automorphism group
556:
502:free abelian group
370:that has solvable
275:
161:
108:with Cayley graph
90:δ-hyperbolic space
65:with respect to a
46:of finite volume.
34:fundamental groups
642:Hyperbolic groups
527:
409:classifying space
384:conjugacy problem
355:convergence group
257:
143:
97:Formal definition
709:
565:
563:
562:
557:
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544:
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529:
528:
520:
469:hyperbolic group
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276:
259:
258:
250:
175:, add a vertex
170:
168:
167:
162:
145:
144:
136:
30:hyperbolic group
717:
716:
712:
711:
710:
708:
707:
706:
692:
691:
667:Jason Behrstock
634:
533:
513:
512:
464:
411:, then so does
325:
243:
242:
129:
128:
99:
52:
12:
11:
5:
715:
713:
705:
704:
694:
693:
690:
689:
680:
673:, Lee Mosher,
671:Cornelia Druţu
664:
657:
645:
638:Mikhail Gromov
633:
630:
629:
628:
617:
586:
583:
580:
577:
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547:
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498:
476:
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417:
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392:
391:
359:
358:
342:
341:
324:
321:
302:
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294:
274:
271:
268:
265:
262:
256:
253:
234:is said to be
211:to the vertex
199:, add an edge
160:
157:
154:
151:
148:
142:
139:
98:
95:
51:
48:
13:
10:
9:
6:
4:
3:
2:
714:
703:
700:
699:
697:
687:
686:
681:
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672:
668:
665:
662:
659:Benson Farb,
658:
655:
654:
649:
646:
643:
639:
636:
635:
631:
626:
622:
618:
615:
611:
607:
604:<5, where
603:
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584:
581:
578:
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545:
540:
510:
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487:cusp subgroup
484:
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452:
451:
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440:
439:
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423:
419:
418:
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407:has a finite
406:
402:
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382:has solvable
381:
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127:
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115:
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107:
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96:
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91:
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83:
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68:
64:
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57:
49:
47:
45:
42:
39:
35:
31:
28:concept of a
27:
23:
19:
684:
675:
660:
652:
641:
613:
605:
601:
597:
574:
570:
508:
504:
471:, such as a
446:
442:
433:
425:
421:
412:
404:
400:
396:
387:
379:
375:
372:word problem
367:
363:
350:
346:
337:
333:
329:
316:
314:
309:
305:
303:
297:
291:δ-hyperbolic
237:
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204:
200:
196:
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188:
184:
180:
176:
172:
125:
121:
117:
113:
109:
105:
100:
81:
77:
74:Cayley graph
69:
62:
58:
53:
21:
15:
420:If a group
395:If a group
362:If a group
345:If a group
328:If a group
18:mathematics
648:Denis Osin
632:References
473:free group
323:Properties
41:noncompact
525:^
522:Γ
415:(Dahmani)
255:^
252:Γ
141:^
138:Γ
696:Category
623:and the
480:complete
462:Examples
228:Bowditch
101:Given a
67:subgroup
38:complete
608:is the
594:surface
432:, then
386:, then
374:, then
336:, then
403:, and
296:it is
289:It is
221:proper
185:Γ
110:Γ
86:cosets
80:along
610:genus
56:group
619:The
612:and
588:The
500:The
467:Any
453:The
298:fine
293:and
195:of
120:of
76:of
61:is
36:of
16:In
698::
669:,
650:,
640:,
312:.
217:gH
197:gH
181:gH
173:gH
54:A
614:n
606:g
602:n
600:+
598:g
575:H
571:H
554:)
550:Z
546:,
541:2
536:Z
531:(
509:Z
505:Z
497:.
447:G
443:H
434:G
426:H
422:G
413:G
405:H
401:H
397:G
388:G
380:H
376:G
368:H
364:G
351:H
347:G
338:G
334:H
330:G
317:G
310:H
306:G
273:)
270:H
267:,
264:G
261:(
238:H
232:G
215:(
213:v
209:x
205:x
203:(
201:e
193:x
189:G
187:(
179:(
177:v
159:)
156:H
153:,
150:G
147:(
122:G
118:H
114:G
112:(
106:G
84:-
82:H
78:G
70:H
59:G
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