Connectivity at infinity for right angled Artin groups

Authors:
Noel Brady and John Meier

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

Published electronically:
August 21, 2000

MathSciNet review:
1675166

Full-text PDF

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]**M. Bestvina,*Nonpositively curved aspects of Artin groups of finite type*, Geometry and Topology**3**(1999), 269-302. MR**2000h:20079****[BB]**M. Bestvina and N. Brady,*Morse theory and finiteness properties of groups*, Invent. Math.**129**(1997), 445-470. MR**98i:20039****[BF]**M. Bestvina and M. Feighn,*The topology at infinity of Out*, Invent. Math.**140**(2000), 651-692.**[BH]**M. Bridson and A. Haefliger,*Metric spaces of non-positive 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, Springer-Verlag, 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), 171-188. CMP**2000:13****[Da 1]**M. W. Davis,*Groups generated by reflections and aspherical manifolds not covered by Euclidean space*, Ann. Math.**117**(1987), 293-324. MR**86d:57025****[Da 2]**M. W. Davis,*The cohomology of a Coxeter group with group ring coefficients*, Duke Math. Jour**91**(1998), 297-314. 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), 123-137. MR**86h:20074****[GM 2]**R. Geoghegan and M. L. Mihalik,*The Fundamental Group at Infinity*, Topology**35**(1996), 655-669. MR**97h:57002****[H]**C. H. Houghton,*Cohomology and the behavior at infinity of finitely presented groups*, J. Lond. Math. Soc. (2)**15**(1977), 465-471. MR**56:15782****[J]**B. Jackson,*End invariants of group extensions*, Topology**21**(1982), 71-81. MR**83a:57002****[MS]**S. Mardesic and J. Segal,*Shape Theory*, North-Holland, (1982). MR**84b:55020****[Me]**J. Meier,*Geometric invariants for Artin groups*, Proc. London Math. Soc. (3)**74**(1997), 151-173. 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), 22-44. MR**99f:57002****[MV]**J. Meier and L. VanWyk,*The Bieri-Neumann-Strebel invariants for graph groups*, Proc. London Math. Soc. (3)**71**(1995), 263-280. MR**96h:20093****[Mi 1]**M. L. Mihalik,*Semistability at the end of a group extension*, Trans. Amer. Math. Soc.**277**(1983), 307-321. MR**84d:57001****[Mi 2]**M. L. Mihalik,*Semistability of Artin and Coxeter groups*, J. Pure Appl. Algebra**111**(1996), 205-211. MR**97e:20060****[Mi 3]**M. L. Mihalik,*Semistability at infinity, simple connectivity at infinity, and normal subgroups*, Top. Appl.**72**(1996), 273-281. 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), 218-232. MR**90k:20057****[R]**J. P. Rickert,*A proof of the simple connectivity at infinity of*, J. Pure Appl. Algebra**145**(2000), 59-73. CMP**2000:06****[St]**J. R. Stallings,*On torsion free groups with infinitely many ends*, Ann. of Math.**88**(1968), 312-334. MR**37:4153****[Sr]**R. Strebel,*A remark on subgroups of infinite index in Poincaré duality groups*, Comment. Math. Helv.**52**(1977), 317-324. MR**56:15793****[V]**K. Vogtmann,*End invariants of the group of outer automorphisms of a free group*, Topology**34**(1995), 533-545. MR**96h:20068**

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:
https://doi.org/10.1090/S0002-9947-00-02506-X

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 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

Article copyright:
© Copyright 2000
American Mathematical Society