485:. The quotient of the simplicial complex has finite stabilizer groups attached to vertices, edges and triangles together with monomorphisms for every inclusion of simplices. A complex of groups is said to be
489:
if it arises as the quotient of a CAT(0) simplicial complex. Developability is a non-positive curvature condition on the complex of groups: it can be verified locally by checking that all
665:
739:
709:
646:
616:
585:
785:
426:. The graph of groups given by the stabilizer subgroups on the fundamental domain corresponds to the original graph of groups.
505:
436:
321:
29:
41:
468:
317:
501:
522:
490:
704:, London Math. Soc. Lecture Note Ser., vol. 36, Cambridge: Cambridge University Press, pp. 137–203,
471:
17:
517:
61:
443:
407:
57:
49:
25:
611:, Cambridge Studies in Advanced Mathematics, vol. 17, Cambridge: Cambridge University Press,
382:. It is possible to define a graph with vertices and edges the disjoint union of all coset spaces
497:
of vertices have length at least six. Such complexes of groups originally arose in the theory of
478:
423:
313:
634:
573:
641:, Progress in Mathematics (in French), vol. 83, Boston, MA: Birkhäuser, pp. 203–213,
735:
727:
705:
642:
612:
581:
69:
674:
547:
749:
719:
686:
656:
626:
595:
561:
745:
715:
682:
652:
622:
604:
591:
580:, Grundlehren der Mathematischen Wissenschaften , vol. 319, Berlin: Springer-Verlag,
557:
494:
754:
693:
569:
320:
for the elements of the fundamental groupoid. This includes normal form theorems for a
53:
779:
697:
552:
447:
325:
170:
482:
45:
33:
504:; their general definition and continued study have been inspired by the ideas of
678:
36:
of the edge groups into the vertex groups. There is a unique group, called the
762:
535:
65:
459:
The simplest possible generalisation of a graph of groups is a 2-dimensional
435:
A graph of groups on a graph with one edge and two vertices corresponds to a
312:, is that it is defined independently of base point or tree. Also there is
464:
460:
305:
663:
Higgins, P. J. (1976), "The fundamental groupoid of a graph of groups",
639:
Sur les groupes hyperboliques d'après
Mikhael Gromov (Bern, 1988)
734:, Springer Monographs in Mathematics, Berlin: Springer-Verlag,
344:
be the fundamental group corresponding to the spanning tree
52:: the original graph of groups can be recovered from the
157:
into the groups assigned to the vertices at its ends.
44:
graph of groups. It admits an orientation-preserving
538:(1993), "Covering theory for graphs of groups",
297:This definition is independent of the choice of
700:(1979), "Topological Methods in Group Theory",
184:to be the group generated by the vertex groups
8:
442:A graph of groups on a single vertex with a
24:is an object consisting of a collection of
666:Journal of the London Mathematical Society
551:
304:The benefit in defining the fundamental
40:, canonically associated to each finite
578:Metric Spaces of Non-Positive Curvature
378:can be identified with their images in
309:
60:. This theory, commonly referred to as
28:indexed by the vertices and edges of a
761:", written with the collaboration of
7:
329:
540:Journal of Pure and Applied Algebra
308:of a graph of groups, as shown by
14:
88:is an assignment to each vertex
474:actions of discrete groups on
437:free product with amalgamation
406:respectively. This graph is a
322:free product with amalgamation
203:with the following relations:
1:
230:with the reverse orientation.
553:10.1016/0022-4049(93)90085-8
32:, together with a family of
757:from "arbres, amalgames, SL
481:that have the structure of
802:
418:acts. It admits the graph
637:(1990), "Orbi-espaces ",
126:as well as monomorphisms
772:(1983). See Chapter I.5.
702:Homological Group Theory
679:10.1112/jlms/s2-13.1.145
523:Right-angled Artin group
64:, is due to the work of
609:Groups Acting on Graphs
463:. These are modeled on
412:universal covering tree
786:Geometric group theory
472:properly discontinuous
18:geometric group theory
502:Bruhat–Tits buildings
479:simplicial complexes
58:stabilizer subgroups
348:. For every vertex
728:Serre, Jean-Pierre
570:Bridson, Martin R.
446:corresponds to an
424:fundamental domain
518:Bass–Serre theory
493:occurring in the
461:complex of groups
336:Structure theorem
199:for each edge of
179:fundamental group
161:Fundamental group
107:and to each edge
70:Jean-Pierre Serre
62:Bass–Serre theory
38:fundamental group
793:
753:. Translated by
752:
722:
689:
659:
635:Haefliger, André
629:
598:
574:Haefliger, André
564:
555:
500:
477:
421:
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393:
381:
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145:
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95:
91:
87:
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795:
794:
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790:
776:
775:
765:, 3rd edition,
760:
742:
726:
712:
692:
662:
649:
633:
619:
605:Dunwoody, M. J.
603:Dicks, Warren;
602:
588:
568:
534:
531:
514:
498:
475:
457:
455:Generalisations
432:
419:
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208:
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196:
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185:
181:
177:and define the
174:
166:
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147:
144:
137:
134:
127:
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116:
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108:
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97:
93:
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82:graph of groups
78:
22:graph of groups
12:
11:
5:
799:
797:
789:
788:
778:
777:
774:
773:
758:
755:John Stillwell
740:
724:
710:
690:
673:(1): 145–149,
669:, 2nd Series,
660:
647:
631:
617:
600:
586:
566:
530:
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510:
456:
453:
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451:
440:
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428:
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388:
372:
361:
337:
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310:Higgins (1976)
295:
294:
289:is an edge in
277:
270:
253:
238:
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189:
162:
159:
151:
139:
129:
120:
101:
77:
74:
54:quotient graph
13:
10:
9:
6:
4:
3:
2:
798:
787:
784:
783:
781:
771:
768:
764:
756:
751:
747:
743:
741:3-540-44237-5
737:
733:
729:
725:
721:
717:
713:
711:0-521-22729-1
707:
703:
699:
695:
691:
688:
684:
680:
676:
672:
668:
667:
661:
658:
654:
650:
648:0-8176-3508-4
644:
640:
636:
632:
628:
624:
620:
618:0-521-23033-0
614:
610:
606:
601:
597:
593:
589:
587:3-540-64324-9
583:
579:
575:
571:
567:
563:
559:
554:
549:
546:(1–2): 3–47,
545:
541:
537:
533:
532:
528:
524:
521:
519:
516:
515:
511:
509:
507:
503:
499:2-dimensional
496:
492:
488:
484:
483:CAT(0) spaces
480:
476:2-dimensional
473:
470:
467:arising from
466:
462:
454:
449:
448:HNN extension
445:
441:
438:
434:
433:
429:
427:
425:
413:
410:, called the
409:
403:
399:
391:
387:
375:
371:
364:
360:
335:
333:
331:
327:
326:HNN extension
323:
319:
316:there a nice
315:
311:
307:
302:
282:
278:
273:
269:
256:
251:
247:
241:
236:
232:
223:
216:
211:
206:
205:
204:
195:and elements
192:
188:
180:
172:
171:spanning tree
160:
158:
154:
150:
142:
132:
123:
119:
104:
100:
84:over a graph
83:
75:
73:
71:
67:
63:
59:
55:
51:
47:
43:
39:
35:
34:monomorphisms
31:
27:
23:
19:
769:
766:
731:
701:
694:Scott, Peter
670:
664:
638:
608:
577:
543:
539:
486:
458:
411:
401:
397:
389:
385:
373:
369:
362:
358:
339:
303:
296:
280:
271:
267:
254:
249:
245:
239:
234:
226:is the edge
221:
214:
209:
190:
186:
178:
164:
152:
148:
140:
130:
121:
117:
102:
98:
81:
79:
37:
21:
15:
698:Wall, Terry
536:Bass, Hyman
487:developable
414:, on which
324:and for an
318:normal form
115:of a group
96:of a group
767:astérisque
763:Hyman Bass
529:References
76:Definition
66:Hyman Bass
469:cocompact
465:orbifolds
352:and edge
330:Bass 1993
42:connected
780:Category
730:(2003),
607:(1989),
576:(1999),
512:See also
491:circuits
430:Examples
306:groupoid
261:for all
146:mapping
56:and the
750:1954121
720:0564422
687:0401927
657:1086659
627:1001965
596:1744486
562:1239551
748:
738:
718:
708:
685:
655:
645:
625:
615:
594:
584:
560:
506:Gromov
314:proved
46:action
26:groups
732:Trees
495:links
169:be a
48:on a
30:graph
736:ISBN
706:ISBN
643:ISBN
613:ISBN
582:ISBN
444:loop
408:tree
394:and
367:and
340:Let
173:for
165:Let
136:and
68:and
50:tree
20:, a
675:doi
548:doi
422:as
332:).
285:if
283:= 1
265:in
259:(x)
252:= φ
218:if
111:of
92:of
16:In
782::
770:46
746:MR
744:,
716:MR
714:,
696:;
683:MR
681:,
671:13
653:MR
651:,
623:MR
621:,
592:MR
590:,
572:;
558:MR
556:,
544:89
542:,
508:.
396:Γ/
384:Γ/
356:,
301:.
257:,1
248:)
242:,0
213:=
143:,1
133:,0
80:A
72:.
759:2
723:.
677::
630:.
599:.
565:.
550::
450:.
439:.
420:Y
416:Γ
402:y
398:G
390:x
386:G
380:Γ
374:y
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363:x
359:G
354:y
350:x
346:T
342:Γ
328:(
299:T
293:.
291:T
287:y
281:y
276:.
272:y
268:G
263:x
255:y
250:y
246:x
244:(
240:y
237:φ
235:y
228:y
222:y
215:y
210:y
201:Y
197:y
191:x
187:G
182:Γ
175:Y
167:T
153:y
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141:y
138:φ
131:y
128:φ
122:y
118:G
113:Y
109:y
103:x
99:G
94:Y
90:x
86:Y
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