Co-contractions of graphs and right-angled Artin groups

Kim, Sang-hyun¹

¹Department of Mathematics, University of Texas, 1 University Station C1200, Austin, TX 78712-0257, USA

Algebraic and Geometric Topology, Tome 8 (2008) no. 2, pp. 849-868

Résumé

We define an operation on finite graphs, called co-contraction. Then we show that for any co-contraction Γ̂ of a finite graph Γ, the right-angled Artin group on Γ contains a subgroup which is isomorphic to the right-angled Artin group on Γ̂. As a corollary, we exhibit a family of graphs, without any induced cycle of length at least 5, such that the right-angled Artin groups on those graphs contain hyperbolic surface groups. This gives the negative answer to a question raised by Gordon, Long and Reid.

Détail

DOI : 10.2140/agt.2008.8.849

Keywords: right-angled Artin group, graph group, co-contraction, surface group

Bibliographie

[1] R Charney, M W Davis, Finite $K(\pi, 1)$s for Artin groups, from: "Prospects in topology (Princeton, NJ, 1994)", Ann. of Math. Stud. 138, Princeton Univ. Press (1995) 110

[2] J Crisp, M Sapir, M Sageev, Surface subgroups of right-angled Artin groups

[3] J Crisp, B Wiest, Embeddings of graph braid and surface groups in right-angled Artin groups and braid groups, Algebr. Geom. Topol. 4 (2004) 439

[4] M W Davis, T Januszkiewicz, Right-angled Artin groups are commensurable with right-angled Coxeter groups, J. Pure Appl. Algebra 153 (2000) 229

[5] C Droms, Graph groups, coherence, and three-manifolds, J. Algebra 106 (1987) 484

[6] C Droms, Isomorphisms of graph groups, Proc. Amer. Math. Soc. 100 (1987) 407

[7] C Droms, Subgroups of graph groups, J. Algebra 110 (1987) 519

[8] G Duchamp, J Y Thibon, Simple orderings for free partially commutative groups, Internat. J. Algebra Comput. 2 (1992) 351

[9] C M Gordon, Artin groups, $3$–manifolds and coherence, Bol. Soc. Mat. Mexicana $(3)$ 10 (2004) 193

[10] C M Gordon, D D Long, A W Reid, Surface subgroups of Coxeter and Artin groups, J. Pure Appl. Algebra 189 (2004) 135

[11] T Hsu, D T Wise, On linear and residual properties of graph products, Michigan Math. J. 46 (1999) 251

[12] S P Humphries, On representations of Artin groups and the Tits conjecture, J. Algebra 169 (1994) 847

[13] K H Kim, L Makar-Limanov, J Neggers, F W Roush, Graph algebras, J. Algebra 64 (1980) 46

[14] Kim,S., Hyperbolic surface subgroups of right-angled \mboxArtin groups and graph products of groups, PhD thesis, Yale University (2007)

[15] J Meier, L Van Wyk, The Bieri–Neumann–Strebel invariants for graph groups, Proc. London Math. Soc. $(3)$ 71 (1995) 263

[16] V Metaftsis, E Raptis, On the profinite topology of right-angled \mboxArtin groups

[17] G A Niblo, L D Reeves, The geometry of cube complexes and the complexity of their fundamental groups, Topology 37 (1998) 621

[18] J P Serre, Trees, Springer Monographs in Math., Springer (2003)

[19] H Servatius, C Droms, B Servatius, Surface subgroups of graph groups, Proc. Amer. Math. Soc. 106 (1989) 573

[20] L Van Wyk, Graph groups are biautomatic, J. Pure Appl. Algebra 94 (1994) 341