## Automorphisms of Coxeter groups

HTML articles powered by AMS MathViewer

- by Patrick Bahls PDF
- Trans. Amer. Math. Soc.
**358**(2006), 1781-1796 Request permission

## Abstract:

We compute $\textrm {Aut}(W)$ for any even Coxeter group whose Coxeter diagram is connected, contains no edges labeled 2, and cannot be separated into more than 2 connected components by removing a single vertex. The description is given explicitly in terms of the given presentation for the Coxeter group and admits an easy characterization of those groups $W$ for which $\textrm {Out}(W)$ is finite.## References

- Bahls, P., “Even rigidity in Coxeter groups”, Ph.D. Thesis, Vanderbilt University, 2002.
- Patrick Bahls,
*Strongly rigid even Coxeter groups*, Topology Proc.**28**(2004), no. 1, 19–54. Spring Topology and Dynamical Systems Conference. MR**2105446** - Patrick Bahls and Michael Mihalik,
*Reflection independence in even Coxeter groups*, Geom. Dedicata**110**(2005), 63–80. MR**2136020**, DOI 10.1007/s10711-003-1134-z - Bahls, P., and Mihalik, M., “Centralizers of parabolic subgroups of Coxeter groups”, preprint, 2003.
- Nicolas Bourbaki,
*Éléments de mathématique*, Masson, Paris, 1981 (French). Groupes et algèbres de Lie. Chapitres 4, 5 et 6. [Lie groups and Lie algebras. Chapters 4, 5 and 6]. MR**647314** - Ruth Charney and Michael Davis,
*When is a Coxeter system determined by its Coxeter group?*, J. London Math. Soc. (2)**61**(2000), no. 2, 441–461. MR**1760693**, DOI 10.1112/S0024610799008583 - W. N. Franzsen and R. B. Howlett,
*Automorphisms of Coxeter groups of rank three*, Proc. Amer. Math. Soc.**129**(2001), no. 9, 2607–2616. MR**1838783**, DOI 10.1090/S0002-9939-01-05878-6 - W. N. Franzsen,
*Automorphisms of Coxeter groups of rank 3 with infinite bonds*, J. Algebra**248**(2002), no. 1, 381–396. MR**1879023**, DOI 10.1006/jabr.2001.9049 - N. D. Gilbert,
*Presentations of the automorphism group of a free product*, Proc. London Math. Soc. (3)**54**(1987), no. 1, 115–140. MR**872253**, DOI 10.1112/plms/s3-54.1.115 - R. B. Howlett, P. J. Rowley, and D. E. Taylor,
*On outer automorphism groups of Coxeter groups*, Manuscripta Math.**93**(1997), no. 4, 499–513. MR**1465894**, DOI 10.1007/BF02677488 - Lynne D. James,
*Complexes and Coxeter groups—operations and outer automorphisms*, J. Algebra**113**(1988), no. 2, 339–345. MR**929764**, DOI 10.1016/0021-8693(88)90163-9 - Levitt, G., “Automorphisms of hyperbolic groups and graphs of groups”, preprint, 2003.
- Roger C. Lyndon and Paul E. Schupp,
*Combinatorial group theory*, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1977 edition. MR**1812024**, DOI 10.1007/978-3-642-61896-3 - Mihalik, M., and Tschantz, S., “Visual decompositions of Coxeter groups”, preprint, 2001.
- Bernhard Mühlherr,
*Automorphisms of graph-universal Coxeter groups*, J. Algebra**200**(1998), no. 2, 629–649. MR**1610676**, DOI 10.1006/jabr.1997.7230 - Bernhard Mühlherr and Richard Weidmann,
*Rigidity of skew-angled Coxeter groups*, Adv. Geom.**2**(2002), no. 4, 391–415. MR**1941338**, DOI 10.1515/advg.2002.018 - E. Rips and Z. Sela,
*Structure and rigidity in hyperbolic groups. I*, Geom. Funct. Anal.**4**(1994), no. 3, 337–371. MR**1274119**, DOI 10.1007/BF01896245 - Jacques Tits,
*Sur le groupe des automorphismes de certains groupes de Coxeter*, J. Algebra**113**(1988), no. 2, 346–357 (French). MR**929765**, DOI 10.1016/0021-8693(88)90164-0

## Additional Information

**Patrick Bahls**- Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
- Email: pbahls@math.uiuc.edu
- Received by editor(s): May 20, 2003
- Received by editor(s) in revised form: July 9, 2004
- Published electronically: October 21, 2005
- Additional Notes: The author was supported by an NSF VIGRE postdoctoral grant.
- © Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**358**(2006), 1781-1796 - MSC (2000): Primary 20F28, 20F55
- DOI: https://doi.org/10.1090/S0002-9947-05-03779-7
- MathSciNet review: 2186996