Connectivity at infinity for right angled Artin groups
Authors:
Noel Brady and John Meier
Journal:
Trans. Amer. Math. Soc. 353 (2001), 117132
MSC (2000):
Primary 20F36, 57M07
Published electronically:
August 21, 2000
MathSciNet review:
1675166
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We establish sufficient conditions implying semistability and connectivity at infinity properties for CAT(0) cubical complexes. We use this, along with the geometry of cubical 's to give a complete description of the higher connectivity at infinity properties of right angled Artin groups. Among other things, this determines which right angled Artin groups are duality groups. Applications to group extensions are also included.
 [B]
Mladen
Bestvina, Nonpositively curved aspects of Artin groups of finite
type, Geom. Topol. 3 (1999), 269–302
(electronic). MR
1714913 (2000h:20079), http://dx.doi.org/10.2140/gt.1999.3.269
 [BB]
Mladen
Bestvina and Noel
Brady, Morse theory and finiteness properties of groups,
Invent. Math. 129 (1997), no. 3, 445–470. MR 1465330
(98i:20039), http://dx.doi.org/10.1007/s002220050168
 [BF]
M. Bestvina and M. Feighn, The topology at infinity of Out, Invent. Math. 140 (2000), 651692.
 [BH]
M. Bridson and A. Haefliger, Metric spaces of nonpositive curvature, manuscript of a book, in progress.
 [BT]
Matthew
G. Brin and T.
L. Thickstun, 3manifolds which are end 1movable, Mem. Amer.
Math. Soc. 81 (1989), no. 411, viii+73. MR 992161
(90g:57015), http://dx.doi.org/10.1090/memo/0411
 [Br]
Kenneth
S. Brown, Cohomology of groups, Graduate Texts in Mathematics,
vol. 87, SpringerVerlag, New YorkBerlin, 1982. MR 672956
(83k:20002)
 [BM]
K.S. Brown and J. Meier, Improper actions and higher connectivity at infinity, Comment. Math. Helv. 75 (2000), 171188. CMP 2000:13
 [Da 1]
Michael
W. Davis, Groups generated by reflections and aspherical manifolds
not covered by Euclidean space, Ann. of Math. (2) 117
(1983), no. 2, 293–324. MR 690848
(86d:57025), http://dx.doi.org/10.2307/2007079
 [Da 2]
Michael
W. Davis, The cohomology of a Coxeter group with group ring
coefficients, Duke Math. J. 91 (1998), no. 2,
297–314. MR 1600586
(99b:20067), http://dx.doi.org/10.1215/S001270949809113X
 [G]
R. Geoghegan, Topological Methods in Group Theory, manuscript of a book, in progress.
 [GM 1]
Ross
Geoghegan and Michael
L. Mihalik, Free abelian cohomology of groups and ends of universal
covers, J. Pure Appl. Algebra 36 (1985), no. 2,
123–137. MR
787167 (86h:20074), http://dx.doi.org/10.1016/00224049(85)900659
 [GM 2]
Ross
Geoghegan and Michael
L. Mihalik, The fundamental group at infinity, Topology
35 (1996), no. 3, 655–669. MR 1396771
(97h:57002), http://dx.doi.org/10.1016/00409383(95)00033X
 [H]
C.
H. Houghton, Cohomology and the behaviour at infinity of finitely
presented groups, J. London Math. Soc. (2) 15 (1977),
no. 3, 465–471. MR 0457577
(56 #15782)
 [J]
Brad
Jackson, End invariants of group extensions, Topology
21 (1982), no. 1, 71–81. MR 630881
(83a:57002), http://dx.doi.org/10.1016/00409383(82)900428
 [MS]
Sibe
Mardešić and Jack
Segal, Shape theory, NorthHolland Mathematical Library,
vol. 26, NorthHolland Publishing Co., AmsterdamNew York, 1982. The
inverse system approach. MR 676973
(84b:55020)
 [Me]
John
Meier, Geometric invariants for Artin groups, Proc. London
Math. Soc. (3) 74 (1997), no. 1, 151–173. MR 1416729
(97h:20049), http://dx.doi.org/10.1112/S0024611597000063
 [MMV]
John
Meier, Holger
Meinert, and Leonard
VanWyk, Higher generation subgroup sets and the Σinvariants
of graph groups, Comment. Math. Helv. 73 (1998),
no. 1, 22–44. MR 1610579
(99f:57002), http://dx.doi.org/10.1007/s000140050044
 [MV]
John
Meier and Leonard
VanWyk, The BieriNeumannStrebel invariants for graph groups,
Proc. London Math. Soc. (3) 71 (1995), no. 2,
263–280. MR 1337468
(96h:20093), http://dx.doi.org/10.1112/plms/s371.2.263
 [Mi 1]
Michael
L. Mihalik, Semistability at the end of a group
extension, Trans. Amer. Math. Soc.
277 (1983), no. 1,
307–321. MR
690054 (84d:57001), http://dx.doi.org/10.1090/S00029947198306900544
 [Mi 2]
Michael
L. Mihalik, Semistability of Artin and Coxeter groups, J. Pure
Appl. Algebra 111 (1996), no. 13, 205–211. MR 1394352
(97e:20060), http://dx.doi.org/10.1016/00224049(95)001174
 [Mi 3]
Michael
L. Mihalik, Semistability at infinity, simple connectivity at
infinity and normal subgroups, Topology Appl. 72
(1996), no. 3, 273–281. MR 1406313
(97j:20035), http://dx.doi.org/10.1016/01668641(96)000296
 [P]
Joseph
S. Profio, Using subnormality to show the simple
connectivity at infinity of a finitely presented group, Trans. Amer. Math. Soc. 320 (1990), no. 1, 281–292. MR 961627
(90k:20057), http://dx.doi.org/10.1090/S0002994719900961627X
 [R]
J. P. Rickert, A proof of the simple connectivity at infinity of , J. Pure Appl. Algebra 145 (2000), 5973. CMP 2000:06
 [St]
John
R. Stallings, On torsionfree groups with infinitely many
ends, Ann. of Math. (2) 88 (1968), 312–334. MR 0228573
(37 #4153)
 [Sr]
R.
Strebel, A remark on subgroups of infinite index in Poincaré
duality groups, Comment. Math. Helv. 52 (1977),
no. 3, 317–324. MR 0457588
(56 #15793)
 [V]
Karen
Vogtmann, End invariants of the group of outer automorphisms of a
free group, Topology 34 (1995), no. 3,
533–545. MR 1341807
(96h:20068), http://dx.doi.org/10.1016/00409383(94)00042J
 [B]
 M. Bestvina, Nonpositively curved aspects of Artin groups of finite type, Geometry and Topology 3 (1999), 269302. MR 2000h:20079
 [BB]
 M. Bestvina and N. Brady, Morse theory and finiteness properties of groups, Invent. Math. 129 (1997), 445470. MR 98i:20039
 [BF]
 M. Bestvina and M. Feighn, The topology at infinity of Out, Invent. Math. 140 (2000), 651692.
 [BH]
 M. Bridson and A. Haefliger, Metric spaces of nonpositive curvature, manuscript of a book, in progress.
 [BT]
 M. G. Brin and T. L. Thickstun, manifolds which are end movable, Mem. Amer. Math. Soc. 81 no. 411 (1989). MR 90g:57015
 [Br]
 K. Brown, Cohomology of Groups, vol. GTM 87, SpringerVerlag, New York, 1982. MR 83k:20002
 [BM]
 K.S. Brown and J. Meier, Improper actions and higher connectivity at infinity, Comment. Math. Helv. 75 (2000), 171188. CMP 2000:13
 [Da 1]
 M. W. Davis, Groups generated by reflections and aspherical manifolds not covered by Euclidean space, Ann. Math. 117 (1987), 293324. MR 86d:57025
 [Da 2]
 M. W. Davis, The cohomology of a Coxeter group with group ring coefficients, Duke Math. Jour 91 (1998), 297314. CMP 99:06; MR 99b:20067
 [G]
 R. Geoghegan, Topological Methods in Group Theory, manuscript of a book, in progress.
 [GM 1]
 R. Geoghegan and M. L. Mihalik, Free abelian cohomology of groups and ends of universal covers, J. Pure Appl. Algebra 36 (1985), 123137. MR 86h:20074
 [GM 2]
 R. Geoghegan and M. L. Mihalik, The Fundamental Group at Infinity, Topology 35 (1996), 655669. MR 97h:57002
 [H]
 C. H. Houghton, Cohomology and the behavior at infinity of finitely presented groups, J. Lond. Math. Soc. (2) 15 (1977), 465471. MR 56:15782
 [J]
 B. Jackson, End invariants of group extensions, Topology 21 (1982), 7181. MR 83a:57002
 [MS]
 S. Mardesic and J. Segal, Shape Theory, NorthHolland, (1982). MR 84b:55020
 [Me]
 J. Meier, Geometric invariants for Artin groups, Proc. London Math. Soc. (3) 74 (1997), 151173. MR 97h:20049
 [MMV]
 J. Meier, H. Meinert and L. VanWyk, Higher generation subgroup sets and the invariants of graph groups, Comment. Math. Helv 73 (1998), 2244. MR 99f:57002
 [MV]
 J. Meier and L. VanWyk, The BieriNeumannStrebel invariants for graph groups, Proc. London Math. Soc. (3) 71 (1995), 263280. MR 96h:20093
 [Mi 1]
 M. L. Mihalik, Semistability at the end of a group extension, Trans. Amer. Math. Soc. 277 (1983), 307321. MR 84d:57001
 [Mi 2]
 M. L. Mihalik, Semistability of Artin and Coxeter groups, J. Pure Appl. Algebra 111 (1996), 205211. MR 97e:20060
 [Mi 3]
 M. L. Mihalik, Semistability at infinity, simple connectivity at infinity, and normal subgroups, Top. Appl. 72 (1996), 273281. MR 97j:20035
 [P]
 J. Profio, Using subnormality to show the simple connectivity at infinity of a finitely presented group, Trans. Amer. Math. Soc. 320 (1990), 218232. MR 90k:20057
 [R]
 J. P. Rickert, A proof of the simple connectivity at infinity of , J. Pure Appl. Algebra 145 (2000), 5973. CMP 2000:06
 [St]
 J. R. Stallings, On torsion free groups with infinitely many ends, Ann. of Math. 88 (1968), 312334. MR 37:4153
 [Sr]
 R. Strebel, A remark on subgroups of infinite index in Poincaré duality groups, Comment. Math. Helv. 52 (1977), 317324. MR 56:15793
 [V]
 K. Vogtmann, End invariants of the group of outer automorphisms of a free group, Topology 34 (1995), 533545. MR 96h:20068
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2000):
20F36,
57M07
Retrieve articles in all journals
with MSC (2000):
20F36,
57M07
Additional Information
Noel Brady
Affiliation:
Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019
Email:
nbrady@math.ou.edu
John Meier
Affiliation:
Department of Mathematics, Lafayette College, Easton, Pennsylvania 18042
Email:
meierj@lafayette.edu
DOI:
http://dx.doi.org/10.1090/S000299470002506X
PII:
S 00029947(00)02506X
Keywords:
Topology at infinity,
right angled Artin groups,
cubical complexes
Received by editor(s):
December 4, 1997
Received by editor(s) in revised form:
February 5, 1999
Published electronically:
August 21, 2000
Additional Notes:
The first author thanks the Universitat Frankfurt for support during the summer of 1997 while part of this work was being carried out. He also acknowledges support from NSF grant DMS9704417. The second author thanks Cornell University for hosting him while on leave from Lafayette College, and the NSF for the support of an RUI grant DMS9705007
Article copyright:
© Copyright 2000
American Mathematical Society
