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10 - Lattices in hyperbolic buildings

Published online by Cambridge University Press:  05 January 2016

Anne Thomas
Affiliation:
University of Sydney
C. S. Aravinda
Affiliation:
TIFR Centre for Applicable Mathematics, Bangalore, India
F. T. Farrell
Affiliation:
Tsinghua University, Beijing
J. -F. Lafont
Affiliation:
Ohio State University
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Print publication year: 2016

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References

[1] P., Abramenko and K. S., Brown, Buildings, vol. 248 of Graduate Texts in Mathematics, Springer, New York, 2008. Theory and applications.Google Scholar
[2] D., Allcock, Infinitely many hyperbolic Coxeter groups through dimension 19, Geom. Topol., 10 (2006), pp. 737–758 (electronic).Google Scholar
[3] E. M., Andreev, Convex polyhedra in Lobačevskiǐ spaces, Mat. Sb. (N.S.), 81 (123) (1970), pp. 445–478.Google Scholar
[4] U., Bader and Y., Shalom, Factor and normal subgroup theorems for lattices in products of groups, Invent. Math., 163 (2006), pp. 415–454.Google Scholar
[5] W., Ballmann and M., Brin, Polygonal complexes and combinatorial group theory, Geom. Dedicata, 50 (1994), pp. 165–191.Google Scholar
[6] W., Ballmann and J., Światkowski, On L2-cohomology and property (T) for automorphism groups of polyhedral cell complexes, Geom. Funct. Anal., 7 (1997), pp. 615–645.Google Scholar
[7] M., Bourdon, Immeubles hyperboliques, dimension conforme et rigidité de Mostow, Geom. Funct. Anal., 7 (1997), pp. 245–268.Google Scholar
[8] M., Bourdon, Sur les immeubles fuchsiens et leur type de quasi-isométrie, Ergodic Theory Dynam. Systems, 20 (2000), pp. 343–364.Google Scholar
[9] M., Bourdon and H., Pajot, Rigidity of quasi-isometries for some hyperbolic buildings, Comment. Math. Helv., 75 (2000), pp. 701–736.Google Scholar
[10] M. R., Bridson and A., Haefliger, Metric spaces of non-positive curvature, vol. 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1999.Google Scholar
[11] K. S., Brown, Buildings, Springer Monographs in Mathematics, Springer-Verlag, New York, 1998. Reprint of the 1989 original.Google Scholar
[12] V. O., Bugaenko, Arithmetic crystallographic groups generated by reflections, and reflective hyperbolic lattices, in Lie groups, their discrete subgroups, and invariant theory, vol. 8 of Adv. Soviet Math., Amer. Math. Soc., Providence, RI, 1992, pp. 33–55.Google Scholar
[13] M., Burger and S., Mozes, CAT(-1)-spaces, divergence groups and their commensurators, J. Amer. Math. Soc., 9 (1996), pp. 57–93.Google Scholar
[14] P.-E., Caprace and B., Rémy, Simplicity and superrigidity of twin building lattices, Invent. Math., 176 (2009), pp. 169–221.Google Scholar
[15] L., Carbone and H., Garland, Existence of lattices in Kac-Moody groups over finite fields, Commun. Contemp. Math., 5 (2003), pp. 813–867.Google Scholar
[16] L., Carbone, R., Kangaslampi, and A., Vdovina, Groups acting simply transitively on hyperbolic buildings, LMS J. Comput. Math. 15 (2012), pp. 101–112.
[17] G., Daskalopoulos, C., Mese, and A., Vdovina, Superrigidity of hyperbolic buildings, Geom. Funct. Anal., 21 (2011), pp. 905–919.Google Scholar
[18] M. W., Davis, Buildings are CAT(0), in Geometry and cohomology in group theory (Durham, 1994), vol. 252 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 1998, pp. 108–123.Google Scholar
[19] M. W., Davis, The geometry and topology of Coxeter groups, vol. 32 of London Mathematical Society Monographs Series, Princeton University Press, Princeton, NJ, 2008.Google Scholar
[20] M. W., Davis, Examples of buildings constructed via covering spaces, Groups Geom. Dyn., 3 (2009), pp. 279–298.Google Scholar
[21] J., Dymara and T., Januszkiewicz, Cohomology of buildings and their automorphism groups, Invent. Math., 150 (2002), pp. 579–627.Google Scholar
[22] J., Dymara and D., Osajda, Boundaries of right-angled hyperbolic buildings, Fund. Math., 197 (2007), pp. 123–165.Google Scholar
[23] F., Esselmann, The classification of compact hyperbolic Coxeter d-polytopes with d + 2 facets, Comment. Math. Helv., 71 (1996), pp. 229–242.Google Scholar
[24] B., Farb, C., Hruska, and A., Thomas, Problems on automorphism groups of nonpositively curved polyhedral complexes and their lattices, in Geometry, rigidity, and group actions, Chicago Lectures in Math., Univ. Chicago Press, Chicago, IL, 2011, pp. 515–560.Google Scholar
[25] A., Felikson and P., Tumarkin, On hyperbolic Coxeter polytopes with mutually intersecting facets, J. Combin. Theory Ser. A, 115 (2008), pp. 121–146.Google Scholar
[26] A., Felikson and P., Tumarkin, Coxeter polytopes with a unique pair of non-intersecting facets, J. Combin. Theory Ser. A, 116 (2009), pp. 875–902.Google Scholar
[27] D., Futer and A., Thomas, Surface quotients of hyperbolic buildings, Int. Math. Res. Not. IMRN 2012, no. 2, 437–477.Google Scholar
[28] D., Gaboriau and F., Paulin, Sur les immeubles hyperboliques, Geom. Dedicata, 88 (2001), pp. 153–197.Google Scholar
[29] G., Gandini, Bounding the homological finiteness length, Bull. Lond. Math. Soc. 44 (2012), pp. 1209–1214.Google Scholar
[30] C., Godsil and G., Royle, Algebraic graph theory, vol. 207 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2001.Google Scholar
[31] R., Gramlich, M., Horn, and B., Mühlherr, Abstract involutions of algebraic groups and of Kac-Moody groups, J. Group Theory, 14 (2011), pp. 213–249.Google Scholar
[32] M., Gromov, Hyperbolic groups, in Essays in group theory, vol. 8 of Math. Sci. Res. Inst. Publ., Springer, New York, 1987, pp. 75–263.Google Scholar
[33] F., Haglund, Commensurability and separability of quasiconvex subgroups, Algebr. Geom. Topol., 6 (2006), pp. 949–1024.Google Scholar
[34] F., Haglund, Finite index subgroups of graph products, Geom. Dedicata, 135 (2008), pp. 167–209.Google Scholar
[35] F., Haglund and F., Paulin, Simplicité de groupes d'automorphismes d'espaces à courbure négative, in The Epstein birthday schrift, vol. 1 of Geom. Topol. Monogr., Geom. Topol. Publ., Coventry, 1998, pp. 181–248 (electronic).Google Scholar
[36] F., Haglund and F., Paulin, Constructions arborescentes d'immeubles, Math. Ann., 325 (2003), pp. 137–164.Google Scholar
[37] S., Hersonsky and F., Paulin, On the rigidity of discrete isometry groups of negatively curved spaces, Comment. Math. Helv., 72 (1997), pp. 349–388.Google Scholar
[38] J. E., Humphreys, Reflection groups and Coxeter groups, vol. 29 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1990.Google Scholar
[39] R., Kangaslampi and A., Vdovina, Cocompact actions on hyperbolic buildings, Internat. J. Algebra Comput., 20 (2010), pp. 591–603.Google Scholar
[40] I. M., Kaplinskaja, The discrete groups that are generated by reflections in the faces of simplicial prisms in Lobačevskii spaces, Mat. Zametki, 15 (1974), pp. 159–164.Google Scholar
[41] I., Kapovich and D. T., Wise, The equivalence of some residual properties of word-hyperbolic groups, J. Algebra, 223 (2000), pp. 562–583.Google Scholar
[42] D. A., Každan, On the connection of the dual space of a group with the structure of its closed subgroups, Funkcional. Anal. i Priložen., 1 (1967), pp. 71–74.Google Scholar
[43] A., Kubena and A., Thomas, Density of commensurators for uniform lattices of right-angled buildings, J. Group Theory 15 (2012), pp. 565–611.Google Scholar
[44] F., Ledrappier and S., Lim, Volume entropy of hyperbolic buildings, J. Mod. Dyn., 4 (2010), pp. 139–165.Google Scholar
[45] E., Leuzinger, Entropy of the geodesic flow for metric spaces and Bruhat-Tits buildings, Adv. Geom., 6 (2006), pp. 475–491.Google Scholar
[46] S., Lim and A., Thomas, Counting overlattices for polyhedral complexes, Topology Proc., 36 (2010), pp. 229–247.Google Scholar
[47] A., Malcev, On isomorphic matrix representations of infinite groups, Rec. Math. [Mat. Sbornik] N.S., 8 (50) (1940), pp. 405–422.Google Scholar
[48] G., Niblo and L., Reeves, Groups acting on CAT(0) cube complexes, Geom. Topol., 1 (1997), p. approx. 7 pp. (electronic).Google Scholar
[49] A. Y., Olshanskiĭ, SQ-universality of hyperbolic groups, Mat. Sb., 186 (1995), pp. 119–132.Google Scholar
[50] M. N., Prokhorov, Absence of discrete groups of reflections with a noncompact fundamental polyhedron of finite volume in a Lobachevskii space of high dimension, Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), pp. 413–424.Google Scholar
[51] B., Rémy, Construction de réseaux en théorie de Kac-Moody, C. R. Acad. Sci. Paris Sér. I Math., 329 (1999), pp. 475–478.Google Scholar
[52] B., Rémy, Topological simplicity, commensurator super-rigidity and nonlinearities of Kac-Moody groups, Geom. Funct. Anal., 14 (2004), pp. 810–852.With an appendix by P. Bonvin.Google Scholar
[53] M., Ronan, Lectures on buildings, University of Chicago Press, Chicago, IL, 2009. Updated and revised.Google Scholar
[54] J., Świątkowski, Trivalent polygonal complexes of nonpositive curvature and Platonic symmetry, Geom. Dedicata, 70 (1998), pp. 87–110.Google Scholar
[55] A., Thomas, Lattices acting on right-angled buildings, Algebr. Geom. Topol., 6 (2006), pp. 1215–1238.Google Scholar
[56] A., Thomas, On the set of covolumes of lattices for Fuchsian buildings, C. R. Math. Acad. Sci. Paris, 344 (2007), pp. 215–218.Google Scholar
[57] A., Thomas, Existence, covolumes and infinite generation of lattices for Davis complexes, Groups Geom. Dyn. 6 (2012), pp. 765–801.Google Scholar
[58] A., Thomas and K., Wortman, Infinite generation of non-cocompact lattices on right-angled buildings, Algebr. Geom. Topol., 11 (2011), pp. 929–938.Google Scholar
[59] W. P., Thurston, Three-dimensional geometry and topology. Vol. 1, vol. 35 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 1997. Edited by Silvio, Levy.Google Scholar
[60] P., Tumarkin, Compact hyperbolic Coxeter n-polytopes with n + 3 facets, Electron. J. Combin., 14 (2007), pp. Research Paper 69, 36 pp. (electronic).Google Scholar
[61] P., Tumarkin and A., Felikson, On bounded hyperbolic d-dimensional Coxeter polytopes with d + 4 hyperfaces, Tr. Mosk. Mat. Obs., 69 (2008), pp. 126–181.Google Scholar
[62] P. V., Tumarkin, Hyperbolic Coxeter polytopes in Hm with n + 2 hyperfacets, Mat. Zametki, 75 (2004), pp. 909–916.Google Scholar
[63] P. V., Tumarkin, Hyperbolic n-dimensional Coxeter polytopes with n + 3 facets, Tr. Mosk. Mat. Obs., 65 (2004), pp. 253–269.Google Scholar
[64] A., Vdovina, Combinatorial structure of some hyperbolic buildings, Math. Z., 241 (2002), pp. 471–478.Google Scholar
[65] È. B., Vinberg, Absence of crystallographic groups of reflections in Lobachevskii spaces of large dimension, Trudy Moskov. Mat. Obshch., 47 (1984), pp. 68–102, 246.Google Scholar
[66] È. B., Vinberg and O. V., Shvartsman, Discrete groups of motions of spaces of constant curvature, in Geometry, II, vol. 29 of Encyclopaedia Math. Sci., Springer, Berlin, 1993, pp. 139–248.Google Scholar
[67] D. T., Wise, The residual finiteness of negatively curved polygons of finite groups, Invent. Math., 149 (2002), pp. 579–617.Google Scholar
[68] X., Xie, Quasi-isometric rigidity of Fuchsian buildings, Topology, 45 (2006), pp. 101–169.Google Scholar
[69] R. J., Zimmer, Ergodic theory and semisimple groups, vol. 81 of Monographs in Mathematics, Birkhäuser Verlag, Basel, 1984.Google Scholar
[70] A., żuk, La propriété (T) de Kazhdan pour les groupes agissant sur les polyèdres, C. R. Acad. Sci. Paris Sér. I Math., 323 (1996), pp. 453–458.Google Scholar

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