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Finite index subgroups of graph products

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Abstract

We prove that every quasiconvex subgroup of a right-angled Coxeter group is an intersection of finite index subgroups. From this we deduce similar separability results for other types of groups, including graph products of finite groups and right-angled Artin groups.

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References

  1. Agol I., Long D.D., Reid A.W.: The Bianchi groups are separable on geometrically finite subgroups. Ann. Math. (2) 153(3), 599–621 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bourdon M.: Immeubles hyperboliques, dimension conforme et rigidité de Mostow. GAFA 7, 245–268 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  3. Burger M., Mozes S.: Lattices in product of trees. Inst. Hautes tudes Sci. Publ. Math. No. 92(2000), 151–194 (2001)

    Google Scholar 

  4. Burns R.G., Karrass A., Solitar D.: A note on groups with separable finitely generated subgroups. Bull. Aust. Math. Soc. 36, 153–160 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chepoi V.: Graphs of some CAT(0) complexes. Adv. Appl. Math. 24(2), 125–179 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  6. Davis M.: Groups generated by reflections and aspherical manifolds not covered by Euclidean space. Ann. Math. (2) 117(2), 293–324 (1983)

    Article  Google Scholar 

  7. Davis, M.: Buildings are CAT(0). In Geometry and Cohomology in Group Theory (Durham 1994). Cambridge UP (1998)

  8. Davis M., Januszkiewicz T.: Right-angled Artin groups are commensurable with right-angled Coxeter groups. J. Pure Appl. Algebra 153, 229–235 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  9. Davis M., Januszkiewicz T., Scott R.: Fundamental groups of blow-ups. Adv. Math. 177(1), 115–179 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Deodhar V.: A note on subgroups generated by reflections in Coxeter groups. Arch. Math. (Basel) 53(6), 543–546 (1989)

    MATH  MathSciNet  Google Scholar 

  11. Droms C.: Subgroups of graph groups. J. Algebra 110(2), 519–522 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dyer M.: Reflection subgroups of Coxeter systems. J. Algebra 135(1), 57–73 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  13. Gitik R.: Doubles of groups and hyperbolic LERF 3-manifolds. Ann. Math. (2) 150(3), 775–806 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  14. Gitik R., Rips E.: On separability properties of groups. Int. J. Algebra Comput. 5, 703–717 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  15. Gerasimov V.: Fixed-point-free actions on cubings. Siberian Adv. Math. 8(3), 36–58 (1998)

    MathSciNet  Google Scholar 

  16. Green, E.: Graph products. Ph.D. Thesis, University of Warwick (1989)

  17. Gromov M.: Hyperbolic groups. In: Gersten, G.M. (ed.) Essays in group theory, Math. Sci. Res. Inst. Publ. (8), pp. 75–263. Springer (1987)

  18. Hall M.: Coset representations in free groups. Trans. Am. Math. Soc. 67, 421–432 (1949)

    Article  MATH  Google Scholar 

  19. Haglund F.: Réseaux de Coxeter-Davis et commensurateurs. Ann. Inst. Fourier (Grenoble) 48(3), 649–666 (1998)

    MATH  MathSciNet  Google Scholar 

  20. Haglund F.: Commensurability and separability of quasiconvex subgroups. Algebr. Geom. Topol. 6, 949–1024 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  21. Haglund, F., Paulin, F.: Simplicité de groupes d’automorphismes d’espaces à courbure négative, The Epstein Birthday Schrift, pp. 181–248 (electronic). Geom. Topol. Monogr., 1, Geom. Topol. Publ., Coventry (1998)

  22. Haglund F., Wise D.: Special Cube Complexes. GAFA 17(5), 1551–1620 (2008)

    Article  MathSciNet  Google Scholar 

  23. Hsu T., Wise D.: On linear and residual properties of graph products. Mich. Math. J. 46(2), 251–259 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  24. Hsu T., Wise D.: Separating quasiconvex subgroups of right-angled Artin groups. Math. Z. 240, 521–548 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  25. Januszkiewicz T., Świa̧tkowski J.: Commensurability of graph products. Algebra Geom. Topol. 1, 587–603 (2001)

    Article  MATH  Google Scholar 

  26. Long D., Reid A.: On subgroup separability in hyperbolic Coxeter groups. Geom. Dedicata 87(1–3), 245–260 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  27. Long D., Niblo G.A.: Subgroup separability and 3-manifold groups. Math. Z. 207, 209–215 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  28. Meier J.: When is the graph product of hyperbolic groups hyperbolic?. Geom. Dedicata 61(1), 29–41 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  29. Noskov N.: Growth of certain non-positively curved cube groups. Euro. J. Combin. 21(5), 659–666 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  30. Sageev M.: Ends of group pairs and non-positively curved cube complexes. Proc. London Math. Soc. (3) 71(3), 585–617 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  31. Scott P.: Subgroups of surface groups are almost geometric. J. London Math. Soc. (2) 17, 555–565 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  32. Short, H.: Quasiconvexity and a theorem of Howson’s. Group theory from a geometrical viewpoint (Trieste, 1990), 168–176, World Sci. Publishing (1991)

  33. Tits J.: Sur le groupe des automorphismes de certains groupes de Coxeter. J. Algebra 113, 346–357 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  34. Wise D.T.: The residual finiteness of negatively curved polygons of finite groups. Invent. Math. 149(3), 579–617 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  35. Wise D.T.: Subgroup separability of the figure 8 knot group. Topology 45(3), 421–463 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Laboratoire de Mathématiques, Université de Paris XI (Paris-Sud), 91405, Orsay, France

    Frédéric Haglund

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  1. Frédéric Haglund
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Correspondence to Frédéric Haglund.

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Haglund, F. Finite index subgroups of graph products. Geom Dedicata 135, 167–209 (2008). https://doi.org/10.1007/s10711-008-9270-0

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