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Asymptotique de la norme $$L^2$$ d’un cycle géodésique dans les revêtements de congruence d’une variété hyperbolique arithmétique. (The asymptotics of the $$L^2$$ norm of a geodesic cycle in the congruence tower above a hyperbolic arithmetic variety). (French) Zbl 1021.53023
The article under review measures the contribution of geodesic cycles in a finite covering of a hyperbolic arithmetic manifold $$M$$, in the sense of S. K. Yeung [Duke Math. J. 73, 201-226 (1994; Zbl 0818.22007)] and W. Lück [Geom. Funct. Anal. 4, 455-481 (1994; Zbl 0853.57021)].
The main result generalises J. J. Millson’s celebrated theorem, on the Betti number of finite coverings of standard compact arithmetic hyperbolic manifolds [Ann. Math. (2) 104, 235-247 (1976; Zbl 0364.53020)]. In the author’s words: “we obtain an explicit value for the growth rate of …the lifts of a codimension one geodesic cycle in the congruence tower above $$M$$”. The article under review is clearly written and provides a new way of calculating non trivial homology classes in certain locally symmetric spaces.

##### MSC:
 53C22 Geodesics in global differential geometry 57R19 Algebraic topology on manifolds and differential topology
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