CAT(-1)-spaces, divergence groups and their commensurators

Burger, M.; Mozes, S.

doi:10.1090/S0894-0347-96-00196-8

CAT(-1)-spaces, divergence groups and their commensurators
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References

N. A’Campo and M. Burger, Réseaux arithmétiques et commensurateurs d’après G.A. Margulis, Invent. Math. 116 (1994), 1–25, .
S. Adams, Boundary amenability for word hyperbolic groups and an application to smooth dynamics of simple groups, Topology 33 (1994), 765–783, .
—, Reduction of cocycles with hyperbolic targets.
Robert Azencott, Espaces de Poisson des groupes localement compacts, Lecture Notes in Mathematics, Vol. 148, Springer-Verlag, Berlin-New York, 1970 (French). MR 0501376, DOI 10.1007/BFb0059352
W. Ballmann, Lectures on spaces of nonpositive curvature.
W. Ballmann, M. Gromov, and V. Schroeder, Manifolds of nonpositive curvature, Progr. in Math., vol. 61, Birkhäuser, Boston, 1985.
W. Ballmann and M. Brin, Polygonal complexes and combinatorial group theory, Geom. Dedicata 50 (1994), no. 2, 165–191. MR 1279883, DOI 10.1007/BF01265309
Hyman Bass, Covering theory for graphs of groups, J. Pure Appl. Algebra 89 (1993), no. 1-2, 3–47. MR 1239551, DOI 10.1016/0022-4049(93)90085-8
Hyman Bass and Ravi Kulkarni, Uniform tree lattices, J. Amer. Math. Soc. 3 (1990), no. 4, 843–902. MR 1065928, DOI 10.1090/S0894-0347-1990-1065928-2
H. Bass and A. Lubotzky, Rigidity of group actions on locally finite trees, Proc. London Math. Soc. 64 (1994), 541–575, .
—, Tree lattices (In preparation).
Nadia Benakli, Polyèdre à géométrie locale donnée, C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), no. 9, 561–564 (French, with English summary). MR 1133484
Nadia Benakli, Polyèdre hyperbolique à groupe d’automorphismes non discret, C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), no. 10, 667–669 (French, with English summary). MR 1135435
I. N. Bernšteĭn and A. V. Zelevinskiĭ, Representations of the group $GL(n,F),$ where $F$ is a local non-Archimedean field, Uspehi Mat. Nauk 31 (1976), no. 3(189), 5–70 (Russian). MR 0425030
Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012, DOI 10.1007/978-1-4612-0941-6
A. Borel and T. A. Springer, Rationality properties of linear algebraic groups. II, Tohoku Math. J. (2) 20 (1968), 443–497. MR 244259, DOI 10.2748/tmj/1178243073
Armand Borel and Jacques Tits, Groupes réductifs, Inst. Hautes Études Sci. Publ. Math. 27 (1965), 55–150 (French). MR 207712, DOI 10.1007/BF02684375
M. Bourdon, Actions quasi-convexes d’un groupe hyperbolique, flot géodésique.
B. H. Bowditch, Some results on the geometry of convex hulls in manifolds of pinched negative curvature, Comment. Math. Helv. 69 (1994), no. 1, 49–81. MR 1259606, DOI 10.1007/BF02564474
M. Bridson and A. Haefliger, Metric spaces of nonpositive curvature (In preparation).
Michel Coornaert, Mesures de Patterson-Sullivan sur le bord d’un espace hyperbolique au sens de Gromov, Pacific J. Math. 159 (1993), no. 2, 241–270 (French, with French summary). MR 1214072, DOI 10.2140/pjm.1993.159.241
M. Coornaert and A. Papadopoulos, Une dichotomie de Hopf pour les flots géodésiques associés aux groupes discrets d’isométries des arbres, Trans. Amer. Math. Soc. 343 (1994), no. 2, 883–898 (French). MR 1207579, DOI 10.1090/S0002-9947-1994-1207579-2
—, Récurrence de marches aléatoires et ergodicité du flot géodésique sur les graphes réguliers, Strasbourg, 1994.
M.W. Davis, Nonpositive curvature and reflection groups.
Harry Furstenberg, A Poisson formula for semi-simple Lie groups, Ann. of Math. (2) 77 (1963), 335–386. MR 146298, DOI 10.2307/1970220
E. Ghys and P. de la Harpe (eds.), Sur les groupes hyperboliques d’après Gromov, Progr. Math., vol. 83, Birkhäuser, Boston, 1990.
M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
F. Haglund, Les polyèdres de Gromov.
E. Hopf, Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung, Ber. Verh. Sächs. Akad. Wiss. Leipzig 91 (1933), 261–304.
Eberhard Hopf, Ergodic theory and the geodesic flow on surfaces of constant negative curvature, Bull. Amer. Math. Soc. 77 (1971), 863–877. MR 284564, DOI 10.1090/S0002-9904-1971-12799-4
V. A. Kaimanovich, Ergodicity of harmonic invariant measures for the geodesic flow on hyperbolic spaces, J. Reine Angew. Math. 455 (1994), 57–103, .
Ying-Sheng Liu, Density of the commensurability groups of uniform tree lattices, J. Algebra 165 (1994), no. 2, 346–359. MR 1273278, DOI 10.1006/jabr.1994.1115
A. Lubotzky, S. Mozes, and R.J. Zimmer, Superrigidity of the commensurability group of tree lattices, Comment. Math. Helv. 69 (1994), 523–548, .
G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17, Springer-Verlag, Berlin, 1991. MR 1090825, DOI 10.1007/978-3-642-51445-6
G. Moussong, Hyperbolic Coxeter groups.
Peter J. Nicholls, The ergodic theory of discrete groups, London Mathematical Society Lecture Note Series, vol. 143, Cambridge University Press, Cambridge, 1989. MR 1041575, DOI 10.1017/CBO9780511600678
S. J. Patterson, The limit set of a Fuchsian group, Acta Math. 136 (1976), no. 3-4, 241–273. MR 450547, DOI 10.1007/BF02392046
Hans Reiter, Classical harmonic analysis and locally compact groups, Clarendon Press, Oxford, 1968. MR 0306811
Jean-Pierre Serre, Trees, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. MR 607504, DOI 10.1007/978-3-642-61856-7
Dennis Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Math. 153 (1984), no. 3-4, 259–277. MR 766265, DOI 10.1007/BF02392379
J. Tits, Algebraic and abstract simple groups, Ann. of Math. (2) 80 (1964), 313–329. MR 164968, DOI 10.2307/1970394
Jacques Tits, Sur le groupe des automorphismes d’un arbre, Essays on topology and related topics (Mémoires dédiés à Georges de Rham), Springer, New York, 1970, pp. 188–211 (French). MR 0299534
N.Th. Varapoulos, Analysis and geometry on groups, Proc. Internat. Congr. Math. (Kyoto 1990), 951–957.
Robert J. Zimmer, Curvature of leaves in amenable foliations, Amer. J. Math. 105 (1983), no. 4, 1011–1022. MR 708371, DOI 10.2307/2374302
Robert J. Zimmer, Amenable ergodic group actions and an application to Poisson boundaries of random walks, J. Functional Analysis 27 (1978), no. 3, 350–372. MR 0473096, DOI 10.1016/0022-1236(78)90013-7
Robert J. Zimmer, Ergodic theory and semisimple groups, Monographs in Mathematics, vol. 81, Birkhäuser Verlag, Basel, 1984. MR 776417, DOI 10.1007/978-1-4684-9488-4