2011/07/19 by Gabriele Link, Link, Gabriele
Mathematics · #20G44 #37F35 #Advanced Algebra and Geometry #Dynamical Systems (math.DS) #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.1107.3755
openalex publication_date 2011/07/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Γ be a discrete group acting by isometries on a product X=X1× X2 of Hadamard spaces. We further require that X1, X2 are locally compact and Γ contains two elements projecting to a pair of independent rank one isometries in each factor. Apart from discrete groups acting by isometries on a product of CAT(-1)-spaces, the probably most interesting examples of such groups are Kac-Moody groups over finite fields acting on the Davis complex of their associated twin building. In a previous article we showed that the regular geometric limit set \Lim splits as a product FΓ× PΓ, where FΓ⊆\rand1× \rand2 is the projection of the geometric limit set to \rand1× \rand2, and PΓ encodes the ratios of the speed of divergence of orbit points in each factor. Our aim in this paper is a description of the limit set from a measure theoretical point of view. We first study the conformal density obtained from the classical Patterson-Sullivan construction, then generalize this construction to obtain measures supported in each Γ-invariant subset of the regular limit set and investigate their properties. Finally we show that the Hausdorff dimension of the radial limit set in each Γ-invariant subset of \Lim is bounded above by the exponential growth rate introduced in the previous article.