# CO-CONTRACTIONS OF GRAPHS AND RIGHT-ANGLED COXETER GROUPS

• Received : 2011.06.03
• Published : 2012.09.30

#### Abstract

We prove that if $\widehat{\Gamma}$ is a co-contraction of ${\Gamma}$, then the right-angled Coxeter group $C(\widehat{\Gamma})$ embeds into $C({\Gamma})$. Further, we provide a graph ${\Gamma}$ without an induced long cycle while $C({\Gamma})$ does not contain a hyperbolic surface group.

#### Acknowledgement

Supported by : Konkuk University

#### References

1. G. Baumslag, Topics in Combinatorial Group Theory, Lectures in Mathematics ETH Zurich, Birkhauser Verlag, Basel, 1993.
2. R. W. Bell. Combinatorial methods for detecting surface subgroups in right-angled Artin groups, arxiv.org/1012.4208.
3. R. Charney, An introduction to right-angled Artin groups, Geom. Dedicata 125 (2007), 141-158. https://doi.org/10.1007/s10711-007-9148-6
4. M. W. Davis and T. Januszkiewicz, Right-angled Artin groups are commensurable with right-angled Coxeter groups, J. Pure Appl. Algebra 153 (2000), no. 3, 229-235. https://doi.org/10.1016/S0022-4049(99)00175-9
5. C. McA. Gordon, D.D. Long, and A. W. Reid, Surface subgroups of Coxeter and Artin groups, J. Pure Appl. Algebra 189 (2004), no. 1-3, 135-148. https://doi.org/10.1016/j.jpaa.2003.10.011
6. J. Gross and T. W. Tucker, Topological Graph Theory, A Wiley-Interscience Publication, John Wiley and Sons, 1987.
7. S. Kim, Hyperbolic Surface Subgroups of Right-Angled Artin Groups and Graph Products of Groups, PhD thesis, Yale University, 2007.
8. S. Kim, Co-contractions of graphs and right-angled Artin groups, Algebr. Geom. Topol. 8 (2008), no. 2, 849-868. https://doi.org/10.2140/agt.2008.8.849
9. R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Springer, Berlin, Heidelberg, New York, 1977.
10. W. Magnus, A. Karrass, and D. Solitar, Combinatorial Group Theory, Dover Publications Inc., 1976.
11. W. Massey, Algebraic Topology: An Introduction, GTM 56, Springer, 1977.
12. P. Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc. (2) 17 (1978), no. 3, 555-565. https://doi.org/10.1112/jlms/s2-17.3.555
13. H. Servatius, C. Droms, and B. Servatius, Surface subgroups of graph group, Proc. Amer. Math. Soc. 106 (1989), no. 3, 573-578. https://doi.org/10.1090/S0002-9939-1989-0952322-9

#### Cited by

1. SURFACE SUBGROUPS OF GRAPH PRODUCTS OF GROUPS vol.22, pp.08, 2012, https://doi.org/10.1142/S0218196712400036