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Benjamini-Schramm convergence of arithmetic orbifolds.

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NPC - Non-positive curvature group actions and cohomology

Let X be the a symmetric space. We say that a sequence of locally symmetric spaces Benjamini-Schramm converges to X if for any real number R the fraction of the volume taken by the R-thin part tends to 0. In my thesis I showed that for a cocompact, congruence arithmetic hyperbolic 3-manifold the volume of the R-thin part is less than a power less than one of the total volume. As a consequence, any sequence of such manifolds Benjamini-Schramm converges to hyperbolic 3-space. I will give some topological applications of this result. Lastly, I will discuss Benjamini-Schramm convergence of congruence arithmetic orbifolds covered by the symmetric spaces of real rank 1.   (joint work with Jean Raimbault).



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