Geometry & Topology

Asymptotic geometry in products of Hadamard spaces with rank one isometries

Gabriele Link

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732212

Digital Object Identifier
doi:10.2140/gt.2010.14.1063

Mathematical Reviews number (MathSciNet)
MR2629900

Zentralblatt MATH identifier
1273.20040

Subjects
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

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

Citation

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. https://projecteuclid.org/euclid.gt/1513732212


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