Geometry & Topology

Asymptotic geometry in products of Hadamard spaces with rank one isometries

Gabriele Link

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In this article we study asymptotic properties of certain discrete groups Γ acting by isometries on a product X=X1×X2 of locally compact Hadamard spaces which admit a geodesic without flat half-plane. The motivation comes from the fact that Kac–Moody groups over finite fields, which can be seen as generalizations of arithmetic groups over function fields, belong to the considered class of groups. Hence one may ask whether classical properties of discrete subgroups of higher rank Lie groups as in Benoist [Geom. Funct. Anal. 7 (1997) 1-47] and Quint [Comment. Math. Helv. 77 (2002) 563-608] hold in this context.

In the first part of the paper we describe the structure of the geometric limit set of Γ and prove statements analogous to the results of Benoist. The second part is concerned with the exponential growth rate δθ(Γ) of orbit points in X with a prescribed “slope” θ(0,π2), which appropriately generalizes the critical exponent in higher rank. In analogy to Quint’s result we show that the homogeneous extension ΨΓ to 02 of δθ(Γ) as a function of θ is upper semicontinuous and concave.

Article information

Geom. Topol., Volume 14, Number 2 (2010), 1063-1094.

Received: 26 June 2009
Revised: 1 March 2010
Accepted: 21 February 2010
First available in Project Euclid: 20 December 2017

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Zentralblatt MATH identifier

Primary: 20F69: Asymptotic properties of groups 51F99: None of the above, but in this section
Secondary: 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 20G15: Linear algebraic groups over arbitrary fields 22D40: Ergodic theory on groups [See also 28Dxx] 51E24: Buildings and the geometry of diagrams

discrete group $\mathrm{CAT}(0)$–spaces limit set critical exponent Kac–Moody groups


Link, Gabriele. Asymptotic geometry in products of Hadamard spaces with rank one isometries. Geom. Topol. 14 (2010), no. 2, 1063--1094. doi:10.2140/gt.2010.14.1063.

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  • W Ballmann, Axial isometries of manifolds of nonpositive curvature, Math. Ann. 259 (1982) 131–144
  • W Ballmann, Lectures on spaces of nonpositive curvature, DMV Seminar 25, Birkhäuser Verlag, Basel (1995) With an appendix by M Brin
  • W Ballmann, M Brin, Orbihedra of nonpositive curvature, Inst. Hautes Études Sci. Publ. Math. (1995) 169–209 (1996)
  • W Ballmann, M Gromov, V Schroeder, Manifolds of nonpositive curvature, Progress in Math. 61, Birkhäuser, Boston (1985)
  • Y Benoist, Propriétés asymptotiques des groupes linéaires, Geom. Funct. Anal. 7 (1997) 1–47
  • M Bestvina, K Fujiwara, A characterization of higher rank symmetric spaces via bounded cohomology, Geom. Funct. Anal. 19 (2009) 11–40
  • M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grund. der Math. Wissenschaften 319, Springer, Berlin (1999)
  • M Burger, Intersection, the Manhattan curve, and Patterson–Sullivan theory in rank $2$, Internat. Math. Res. Notices (1993) 217–225
  • M Burger, N Monod, Continuous bounded cohomology and applications to rigidity theory, Geom. Funct. Anal. 12 (2002) 219–280
  • P-E Caprace, K Fujiwara, Rank one isometries of buildings and quasi-morphisms of Kac–Moody groups
  • P-E Caprace, B Rémy, Simplicity and superrigidity of twin building lattices, Invent. Math. 176 (2009) 169–221
  • F Dal'Bo, Remarques sur le spectre des longueurs d'une surface et comptages, Bol. Soc. Brasil. Mat. $($N.S.$)$ 30 (1999) 199–221
  • F Dal'Bo, I Kim, Shadow lemma on the product of Hadamard manifolds and applications, from: “Actes du Séminaire de Théorie Spectrale et Géométrie. Vol. 25. Année 2006–2007”, Sémin. Théor. Spectr. Géom. 25, Univ. Grenoble I, Saint (2008) 105–119
  • V A Kaimanovich, Double ergodicity of the Poisson boundary and applications to bounded cohomology, Geom. Funct. Anal. 13 (2003) 852–861
  • G Link, Hausdorff dimension of limit sets of discrete subgroups of higher rank Lie groups, Geom. Funct. Anal. 14 (2004) 400–432
  • J-F Quint, Divergence exponentielle des sous-groupes discrets en rang supérieur, Comment. Math. Helv. 77 (2002) 563–608
  • J-F Quint, Mesures de Patterson–Sullivan en rang supérieur, Geom. Funct. Anal. 12 (2002) 776–809
  • B Rémy, Construction de réseaux en théorie de Kac–Moody, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999) 475–478
  • C Yue, The ergodic theory of discrete isometry groups on manifolds of variable negative curvature, Trans. Amer. Math. Soc. 348 (1996) 4965–5005