Innovations in Incidence Geometry

The twist conjecture for Coxeter groups without small triangle subgroups

Christian J. Weigel

Full-text: Open access

Abstract

We prove Mühlherr's twist conjecture for Coxeter systems $(W,S)$ which have no rank $3$ subsystems of type $2$-$3$-$n$ or $2$-$4$-$n$ ($n \geq 3$). In combination with known results this finishes the solution of the isomorphism problem for this class of groups. The condition on the diagram does not allow spherical rank $3$ subsystems, but our result covers “most” of the even Coxeter systems. With respect to earlier contributions, we develop a geometric technique to handle rank $2$ twists, in particular rotation twists which occur in the even case.

Article information

Source
Innov. Incidence Geom., Volume 12, Number 1 (2011), 111-140.

Dates
Received: 19 August 2010
Accepted: 12 January 2011
First available in Project Euclid: 28 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.iig/1551323071

Digital Object Identifier
doi:10.2140/iig.2011.12.111

Mathematical Reviews number (MathSciNet)
MR2942720

Zentralblatt MATH identifier
1284.20042

Subjects
Primary: 20F55: Reflection and Coxeter groups [See also 22E40, 51F15] 51F15: Reflection groups, reflection geometries [See also 20H10, 20H15; for Coxeter groups, see 20F55]

Keywords
Coxeter group twist conjecture

Citation

Weigel, Christian J. The twist conjecture for Coxeter groups without small triangle subgroups. Innov. Incidence Geom. 12 (2011), no. 1, 111--140. doi:10.2140/iig.2011.12.111. https://projecteuclid.org/euclid.iig/1551323071


Export citation

References

  • N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968.
  • N. Brady, J. P. McCammond, B. Mühlherr and W. D. Neumann, Rigidity of Coxeter groups and Artin groups, in Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000), vol. 94, 2002, pp. 91–109.
  • P.-E. Caprace and B. Mühlherr, Reflection rigidity of $2$-spherical Coxeter groups, Proc. Lond. Math. Soc. (3) 94 (2007), no. 2, 520–542.
  • P.-E. Caprace and P. Prztycki, Twist-rigid coxeter groups (2009), arXiv:0911.0354v1 [math.GR], Preprint.
  • 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.
  • B. Mühlherr, Automorphisms of graph-universal Coxeter groups, J. Algebra 200 (1998), no. 2, 629–649.
  • ––––, The isomorphism problem for Coxeter groups, in The Coxeter legacy, Amer. Math. Soc., Providence, RI, 2006, pp. 1–15.
  • B. Mühlherr and R. Weidmann, Rigidity of skew-angled Coxeter groups, Adv. Geom. 2 (2002), no. 4, 391–415.
  • J. G. Ratcliffe and S. T. Tschantz, Chordal Coxeter groups, Geom. Dedicata 136 (2008), 57–77.
  • M. Ronan, Lectures on buildings, University of Chicago Press, Chicago, IL, 2009, updated and revised.