Abstract
For a rank one Lie group G and a Zariski dense and geometrically finite subgroup \({\Gamma}\) 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 \({G={\rm PSL}_2 (\mathbb{C})}\) , our results imply the equidistribution of eigenvalues of elements of Γ in the complex plane.
When \({\Gamma}\) is a lattice, the equidistribution of holonomies was proved by Sarnak and Wakayama in 1999 using the Selberg trace formula.
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Margulis was supported in part by NSF Grant #1265695. Mohammadi was supported in part by NSF Grant #1200388 and Sloan fellowship. Oh was supported in part by NSF Grant #1068094 and #1361673.
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Margulis, G., Mohammadi, A. & Oh, H. Closed geodesics and holonomies for Kleinian manifolds. Geom. Funct. Anal. 24, 1608–1636 (2014). https://doi.org/10.1007/s00039-014-0299-y
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DOI: https://doi.org/10.1007/s00039-014-0299-y