Automorphisms of Coxeter groups

Bahls, Patrick

doi:10.1090/S0002-9947-05-03779-7

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