Generalized conformal densities for higher products of rank one Hadamard spaces
- 72 Downloads
Let \(X\) be a product of locally compact rank one Hadamard spaces and \(\Gamma \) a discrete group of isometries which contains two elements projecting to a pair of independent rank one isometries in each factor. In Link (Asymptotic geometry in higher products of rank one Hadamard spaces. arXiv:1308.5584, 2013) we gave a precise description of the structure of the geometric limit set \(L_\Gamma \) of \(\Gamma \); our aim in this paper is to describe this set from a measure theoretical point of view, using as a basic tool the properties of the exponent of growth of \(\Gamma \) established in the aforementioned article. We first show that the conformal density obtained from the classical Patterson–Sullivan construction is supported in a unique \(\Gamma \)-invariant subset of the geometric limit set; generalizing this classical construction we then obtain measures supported in each \(\Gamma \)-invariant subset of the regular limit set and investigate their properties. We remark that apart from Kac–Moody groups over finite fields acting on the Davis complex of their associated twin building, the probably most interesting examples to which our results apply are isometry groups of reducible CAT(0)-cube complexes without Euclidean factors.
KeywordsCAT (0)-spaces Products Cubical complexes Discrete groups Rank one isometries Limit set Patterson–Sullivan theory
Mathematics Subject Classification20F69 22D40 22G44 51F99
The first draft of this paper was initiated during the author’s stay at IHES in Bures-sur-Yvette. She warmly thanks the institute for its hospitality and the inspiring atmosphere.
- 4.Ballmann, W., Brin, M.: Orbihedra of nonpositive curvature. Inst. Hautes Études Sci. Publ. Math. 82, 169–209 (1995)Google Scholar
- 6.Bridson, M.R., Haefliger, A.: Metric Spaces of Non-positive Curvature, Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 319. Springer, Berlin (1999)Google Scholar
- 11.Dal’Bo, F., Kim, I.: Shadow lemma on the product of Hadamard manifolds and applications. Actes du Séminaire de Théorie Spectrale et Géométrie, vol. 25. Année 2006–2007, Sémin. Théor. Spectr. Géom., vol. 25, Univ. Grenoble I, Saint, 2008, pp. 105–119Google Scholar
- 16.Link, G.: Asymptotic Geometry in Higher Products of Rank One Hadamard Spaces. arXiv:1308.5584 (2013)