Algebraic & Geometric Topology

Angle-deformations in Coxeter groups

Timothée Marquis and Bernhard Mühlherr

Full-text: Open access

Abstract

The isomorphism problem for Coxeter groups has been reduced to its “reflection preserving version” by B Howlett and the second author. Thus, in order to solve it, it suffices to determine for a given Coxeter system (W,R) all Coxeter generating sets S of W which are contained in RW, the set of reflections of (W,R). In this paper, we provide a further reduction: it suffices to determine all Coxeter generating sets SRW which are sharp-angled with respect to R.

Added 22 May 2012: The description of Figure 3 in Section 8.1 has been corrected.

Article information

Source
Algebr. Geom. Topol., Volume 8, Number 4 (2008), 2175-2208.

Dates
Received: 21 February 2008
Revised: 19 October 2008
Accepted: 29 October 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796931

Digital Object Identifier
doi:10.2140/agt.2008.8.2175

Mathematical Reviews number (MathSciNet)
MR2465738

Zentralblatt MATH identifier
1184.20032

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

Keywords
angle-deformation Coxeter group isomorphism problem sharp-angled

Citation

Marquis, Timothée; Mühlherr, Bernhard. Angle-deformations in Coxeter groups. Algebr. Geom. Topol. 8 (2008), no. 4, 2175--2208. doi:10.2140/agt.2008.8.2175. https://projecteuclid.org/euclid.agt/1513796931


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 1337, Hermann, Paris (1968)
  • N Brady, J P McCammond, B Mühlherr, W D Neumann, Rigidity of Coxeter groups and Artin groups, from: “Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000)”, (L Mosher, M Sageev, editors), Geom. Dedicata 94 (2002) 91–109
  • B Brink, R B Howlett, A finiteness property and an automatic structure for Coxeter groups, Math. Ann. 296 (1993) 179–190
  • P-E Caprace, B Mühlherr, Reflection rigidity of $2$–spherical Coxeter groups, Proc. Lond. Math. Soc. $(3)$ 94 (2007) 520–542
  • R Charney, M Davis, When is a Coxeter system determined by its Coxeter group?, J. London Math. Soc. $(2)$ 61 (2000) 441–461
  • V V Deodhar, A note on subgroups generated by reflections in Coxeter groups, Arch. Math. $($Basel$)$ 53 (1989) 543–546
  • M Dyer, Reflection subgroups of Coxeter systems, J. Algebra 135 (1990) 57–73
  • W N Franzsen, R B Howlett, Automorphisms of nearly finite Coxeter groups, Adv. Geom. 3 (2003) 301–338
  • M Grassi, The isomorphism problem for a class of finitely generated Coxeter groups, PhD thesis, University of Sydney (2007)
  • J E Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Math. 29, Cambridge University Press (1990)
  • B Mühlherr, The isomorphism problem for Coxeter groups, from: “The Coxeter legacy”, (C Davis, E W Ellers, editors), Amer. Math. Soc. (2006) 1–15
  • J Ratcliffe, S Tschantz, Chordal Coxeter groups, Geom. Dedicata 136 (2008) 57–77