Connectivity at infinity for right angled Artin groups
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- by Noel Brady and John Meier PDF
- Trans. Amer. Math. Soc. 353 (2001), 117-132 Request permission
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 $K(\pi ,1)$’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.References
- Mladen Bestvina, Non-positively curved aspects of Artin groups of finite type, Geom. Topol. 3 (1999), 269–302. MR 1714913, DOI 10.2140/gt.1999.3.269
- Mladen Bestvina and Noel Brady, Morse theory and finiteness properties of groups, Invent. Math. 129 (1997), no. 3, 445–470. MR 1465330, DOI 10.1007/s002220050168
- M. Bestvina and M. Feighn, The topology at infinity of Out$(F_{n})$, Invent. Math. 140 (2000), 651–692.
- M. Bridson and A. Haefliger, Metric spaces of non-positive curvature, manuscript of a book, in progress.
- Matthew G. Brin and T. L. Thickstun, $3$-manifolds which are end $1$-movable, Mem. Amer. Math. Soc. 81 (1989), no. 411, viii+73. MR 992161, DOI 10.1090/memo/0411
- Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956
- K.S. Brown and J. Meier, Improper actions and higher connectivity at infinity, Comment. Math. Helv. 75 (2000), 171–188.
- 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, DOI 10.2307/2007079
- Michael W. Davis, The cohomology of a Coxeter group with group ring coefficients, Duke Math. J. 91 (1998), no. 2, 297–314. MR 1600586, DOI 10.1215/S0012-7094-98-09113-X
- R. Geoghegan, Topological Methods in Group Theory, manuscript of a book, in progress.
- 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, DOI 10.1016/0022-4049(85)90065-9
- Ross Geoghegan and Michael L. Mihalik, The fundamental group at infinity, Topology 35 (1996), no. 3, 655–669. MR 1396771, DOI 10.1016/0040-9383(95)00033-X
- 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 457577, DOI 10.1112/jlms/s2-15.3.465
- Brad Jackson, End invariants of group extensions, Topology 21 (1982), no. 1, 71–81. MR 630881, DOI 10.1016/0040-9383(82)90042-8
- Sibe Mardešić and Jack Segal, Shape theory, North-Holland Mathematical Library, vol. 26, North-Holland Publishing Co., Amsterdam-New York, 1982. The inverse system approach. MR 676973
- John Meier, Geometric invariants for Artin groups, Proc. London Math. Soc. (3) 74 (1997), no. 1, 151–173. MR 1416729, DOI 10.1112/S0024611597000063
- John Meier, Holger Meinert, and Leonard VanWyk, Higher generation subgroup sets and the $\Sigma$-invariants of graph groups, Comment. Math. Helv. 73 (1998), no. 1, 22–44. MR 1610579, DOI 10.1007/s000140050044
- John Meier and Leonard VanWyk, The Bieri-Neumann-Strebel invariants for graph groups, Proc. London Math. Soc. (3) 71 (1995), no. 2, 263–280. MR 1337468, DOI 10.1112/plms/s3-71.2.263
- Michael L. Mihalik, Semistability at the end of a group extension, Trans. Amer. Math. Soc. 277 (1983), no. 1, 307–321. MR 690054, DOI 10.1090/S0002-9947-1983-0690054-4
- Michael L. Mihalik, Semistability of Artin and Coxeter groups, J. Pure Appl. Algebra 111 (1996), no. 1-3, 205–211. MR 1394352, DOI 10.1016/0022-4049(95)00117-4
- Michael L. Mihalik, Semistability at infinity, simple connectivity at infinity and normal subgroups, Topology Appl. 72 (1996), no. 3, 273–281. MR 1406313, DOI 10.1016/0166-8641(96)00029-6
- 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, DOI 10.1090/S0002-9947-1990-0961627-X
- J. P. Rickert, A proof of the simple connectivity at infinity of $\textit {Out}(F_4)$, J. Pure Appl. Algebra 145 (2000), 59–73.
- John R. Stallings, On torsion-free groups with infinitely many ends, Ann. of Math. (2) 88 (1968), 312–334. MR 228573, DOI 10.2307/1970577
- R. Strebel, A remark on subgroups of infinite index in Poincaré duality groups, Comment. Math. Helv. 52 (1977), no. 3, 317–324. MR 457588, DOI 10.1007/BF02567371
- Karen Vogtmann, End invariants of the group of outer automorphisms of a free group, Topology 34 (1995), no. 3, 533–545. MR 1341807, DOI 10.1016/0040-9383(94)00042-J
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
- 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 DMS-9704417. 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 DMS-9705007
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 353 (2001), 117-132
- MSC (2000): Primary 20F36, 57M07
- DOI: https://doi.org/10.1090/S0002-9947-00-02506-X
- MathSciNet review: 1675166