Hyperbolic manifolds,amalgamated products and critical exponents

We give a new proof of a result due to Y. Shalom: if the fundamental group of a compact real hyperbolic manifold of dimn is a free product of its subgroups A and B over the amalgamated subgroup C, then the critical exponent of C is not smaller than n?2. The proof, which is geometric, allows one to treat the equality case and an extension to variable curvature. To cite this article: G. Besson et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).