Rigidity of amalgamated product in negative curvature
  Let $\Gamma$ be the fundamental group of a compact n-dimensional riemannian manifold X of sectional curvature bounded above by -1. We suppose that $\Gamma$ is a free product of its subgroup A and B over the amalgamated subgroup C. We prove that the critical exponent $\delta(C)$ of C satisfies $\delta(C) \geq n-2$. The equality happens if and only if there exist an embedded compact hypersurface Y in X, totally geodesic, of constant sectional curvature -1, with fundamental group C and which separates X in two connected components whose fundamental groups are A and B. Similar results hold if $\Gamma$ is an HNN extension, or more generally if $\Gamma$ acts on a simplicial tree without fixed point. 
Gerard Besson (IF), Gilles Courtois (CMLS), Sylvain Gallot (IF)