Conjugacy of 2–spherical subgroups of Coxeter groups and parallel walls

Caprace, Pierre-Emmanuel¹

¹Département de Mathématiques, Université Libre de Bruxelles, CP216, Bd du Triomphe, 1050 Bruxelles, Belgium

Algebraic and Geometric Topology, Tome 6 (2006) no. 4, pp. 1987-2029

Résumé

Let (W,S) be a Coxeter system of finite rank (ie |S| is finite) and let A be the associated Coxeter (or Davis) complex. We study chains of pairwise parallel walls in A using Tits’ bilinear form associated to the standard root system of (W,S). As an application, we prove the strong parallel wall conjecture of G Niblo and L Reeves [J Group Theory 6 (2003) 399–413]. This allows to prove finiteness of the number of conjugacy classes of certain one-ended subgroups of W, which yields in turn the determination of all co-Hopfian Coxeter groups of 2–spherical type.

Détail

DOI : 10.2140/agt.2006.6.1987

Keywords: Coxeter group, conjugacy class, Hopfian group, hyperbolic triangle, parallel walls

Bibliographie

[1] R Benedetti, C Petronio, Lectures on hyperbolic geometry, Universitext, Springer (1992)

[2] N Bourbaki, Éléments de mathématique: Groupes et algèbres de Lie. Chapitres 4, 5 et 6, Masson (1981) 290

[3] B Brink, R B Howlett, A finiteness property and an automatic structure for Coxeter groups, Math. Ann. 296 (1993) 179

[4] P E Caprace, F Haglund, On geometric flats in the $\mathrm{CAT}(0)$ realization of Coxeter groups and Tits buildings

[5] P E Caprace, B Mühlherr, Reflection triangles in Coxeter groups and biautomaticity, J. Group Theory 8 (2005) 467

[6] M W Davis, Buildings are $\mathrm{CAT}(0)$, from: "Geometry and cohomology in group theory (Durham, 1994)", London Math. Soc. Lecture Note Ser. 252, Cambridge Univ. Press (1998) 108

[7] T Delzant, Sous-groupes à deux générateurs des groupes hyperboliques, from: "Group theory from a geometrical viewpoint (Trieste, 1990)", World Sci. Publ., River Edge, NJ (1991) 177

[8] V V Deodhar, On the root system of a Coxeter group, Comm. Algebra 10 (1982) 611

[9] V V Deodhar, A note on subgroups generated by reflections in Coxeter groups, Arch. Math. $($Basel$)$ 53 (1989) 543

[10] M Dyer, Reflection subgroups of Coxeter systems, J. Algebra 135 (1990) 57

[11] A A Felikson, Coxeter decompositions of hyperbolic polygons, European J. Combin. 19 (1998) 801

[12] M Gromov, Hyperbolic groups, from: "Essays in group theory", Math. Sci. Res. Inst. Publ. 8, Springer (1987) 75

[13] F Haglund, D Wise, Coxeter cubings are special

[14] R B Howlett, P J Rowley, D E Taylor, On outer automorphism groups of Coxeter groups, Manuscripta Math. 93 (1997) 499

[15] I Kapovich, R Weidmann, Freely indecomposable groups acting on hyperbolic spaces, Internat. J. Algebra Comput. 14 (2004) 115

[16] D Krammer, The conjugacy problem for Coxeter groups, PhD thesis, Universiteit Utrecht (1994)

[17] G A Margulis, È B Vinberg, Some linear groups virtually having a free quotient, J. Lie Theory 10 (2000) 171

[18] G A Niblo, L D Reeves, Coxeter groups act on $\mathrm{CAT}(0)$ cube complexes, J. Group Theory 6 (2003) 399

[19] E Rips, Z Sela, Structure and rigidity in hyperbolic groups I, Geom. Funct. Anal. 4 (1994) 337

[20] M Sageev, Ends of group pairs and non-positively curved cube complexes, Proc. London Math. Soc. $(3)$ 71 (1995) 585

[21] J P Serre, Arbres, amalgames, $\mathrm{SL}_{2}$, Astérisque 46, Société Mathématique de France (1977)