Rigidity of amalgamated product in negative curvature

Besson, Gerard; Courtois, Gilles; Gallot, Sylvain

Rigidity of amalgamated product in negative curvature

Authors

Besson, Gerard

Courtois, Gilles

Gallot, Sylvain

Description

Let $Γ$ be the fundamental group of a compact n-dimensional riemannian manifold X of sectional curvature bounded above by -1. We suppose that $Γ$ is a free product of its subgroup A and B over the amalgamated subgroup C. We prove that the critical exponent $δ(C)$ of C satisfies $δ(C) \geq n-2$. 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 $Γ$ is an HNN extension, or more generally if $Γ$ acts on a simplicial tree without fixed point.