Closed geodesics and holonomies for Kleinian manifolds

By Gregory Margulis, A. Mohammadi, H. Oh. Geometric and Functional Analysis, 2014

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For a rank one Lie group G and a Zariski dense and geometrically finite subgroup Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Gamma}$$\end{document} of G, we establish the joint equidistribution of closed geodesics and their holonomy classes for the associated locally symmetric space. Our result is given in a quantitative form for geometrically finite real hyperbolic manifolds whose critical exponents are big enough. In the case when G=PSL2(C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${G={\rm PSL}_2 (\mathbb{C})}$$\end{document} , our results imply the equidistribution of eigenvalues of elements of Γ in the complex plane. When Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Gamma}$$\end{document} is a lattice, the equidistribution of holonomies was proved by Sarnak and Wakayama in 1999 using the Selberg trace formula.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ee9fc198f4788a229afa02e5f16ab98bdc087496