Abstract
We show that twin building lattices are undistorted in their ambient group; equivalently, the orbit map of the lattice to the product of the associated twin buildings is a quasi-isometric embedding. As a consequence, we provide an estimate of the quasi-flat rank of these lattices, which implies that there are infinitely many quasi-isometry classes of finitely presented simple groups. Finally, we describe how non-distortion of lattices is related to the integrability of the structural cocycle.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abramenko, P., Brown, K.S.: Buildings, Graduate Texts in Mathematics, vol. 248. Springer, New York, Theory and applications (2008)
Burillo J., Cleary S., Stein M.I.: Metrics and embeddings of generalizations of Thompson’s group F. Trans. Amer. Math. Soc. 353(4), 1677–1689 (2001) (electronic)
Bridson M.R., Haefliger A.: Metric Spaces of Non-Positive Curvature, Grundlehren der Mathematischen Wissenschaften, vol. 319. Springer, New York (1999)
Burger M., Mozes S.: Lattices in product of trees. Inst. Hautes Études Sci. Publ. Math. 92, 151–194 (2001)
Bourbaki N.: Éléments de Mathématique. Topologie générale. Chapitres 1 à 4. Springer, New York (2007)
Brin M.G.: Higher-dimensional Thompson groups. Geom. Dedicata 108, 163–192 (2004)
Brown, K.S.: The geometry of finitely presented infinite simple groups, Algorithms and classification in combinatorial group theory (Berkeley, CA, 1989). Math. Sci. Res. Inst. Publ., vol. 23, pp. 121–136. Springer, New York (1992)
Baumgartner U., Rémy B., Willis G.A.: Flat rank of automorphism groups of buildings. Transformation Groups 12, 413–436 (2007)
Carbone L., Garland H.: Lattices in Kac–Moody groups. Math. Res. Lett. 6, 439–448 (1999)
Caprace P.-E., Haglund F.: On geometrical flats in the CAT(0)-realization of Coxeter groups and Tits buildings. Canad. J. Math. 61, 740–761 (2009)
Caprace, P.-E., Monod, N.: Isometry groups of non-positively curved spaces: discrete subgroups, Preprint, to appear in J. Topology (doi:10.1112/jtopol/jtp027) (2008)
Caprace P.-E., Rémy B.: Simplicity and superrigidity of twin building lattices. Invent. Math. 176(1), 169–221 (2009)
Dymara, J., Schick, T.: Buildings have finite asymptotic dimension, Preprint, arXiv:math/ 0703199v1 (2007)
Gelander T., Karlsson A., Margulis G.A.: Superrigidity, generalized harmonic maps and uniformly convex spaces. Geom. Funct. Anal. 17, 1524–1550 (2008)
Gramlich R., Mühlherr B.: Lattices from involutions of Kac–Moody groups. Oberwolfach Rep. 5, 139–140 (2008)
Higman, G.: Finitely presented infinite simple groups, Department of Pure Mathematics, Department of Mathematics, I.A.S. Australian National University, Canberra, 1974, Notes on Pure Mathematics, No. 8 (1974)
Kleiner B.: The local structure of length spaces with curvature bounded above. Math. Z. 231, 409–456 (1999)
Krammer D.: The conjugacy problem for Coxeter groups. Groups Geom. Dyn. 3(1), 71–171 (2009)
Lubotzky A., Mozes S., Raghunathan M.S.: The word and Riemannian metrics on lattices of semisimple groups. Inst. Hautes Études Sci. Publ. Math. 91, 5–53 (2001)
Margulis G.A.: Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 17,(3). Springer, New york (1991)
Monod N.: Continuous bounded cohomology of locally compact groups, Lecture Notes in Mathematics, vol. 1758. Springer, New York (2001)
Monod N.: Superrigidity for irreducible lattices and geometric splitting. J. Amer. Math. Soc. 19, 781–814 (2006)
Papasoglu P.: Homogeneous trees are bi-Lipschitz equivalent. Geom. Dedicata 54, 301–306 (1995)
Rémy B.: Construction de réseaux en théorie de Kac–Moody. C. R. Acad. Sc. Paris 329, 475–478 (1999)
Rémy B.: Integrability of induction cocycles for Kac–Moody groups. Math. Ann. 333, 29–43 (2005)
Röver Claas E.: Constructing finitely presented simple groups that contain Grigorchuk groups. J. Algebra 220(1), 284–313 (1999)
Scott, E.A.: A tour around finitely presented infinite simple groups, Algorithms and classification in combinatorial group theory (Berkeley, CA, 1989). Math. Sci. Res. Inst. Publ., vol. 23, pp. 83–119. Springer, New York (1992)
Serre, J.-P.: Cohomologie des groupes discrets, Prospects in mathematics (Proc. Sympos., Princeton Univ., Princeton, N.J., 1970), Princeton Univ. Press, Princeton, N.J., 1971, pp. 77–169. Ann. of Math. Studies, No. 70
Shalom Y.: Rigidity of commensurators and irreducible lattices. Invent. Math. 141, 1–54 (2000)
Tits J.: Théorie des groupes. Ann. Collège France 89(1988/89), 81–96 (1990)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the Fund for Scientific Research–F.N.R.S., Belgium.
Supported in part by ANR project GGPG: Géométrie et Probabilités dans les Groupes.
Rights and permissions
About this article
Cite this article
Caprace, PE., Rémy, B. Non-distortion of twin building lattices. Geom Dedicata 147, 397–408 (2010). https://doi.org/10.1007/s10711-010-9469-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-010-9469-8