Besson, Gérard; Courtois, Gilles; Gallot, Sylvain Rigidity of amalgamated products in negative curvature. (English) Zbl 1206.53038 J. Differ. Geom. 79, No. 3, 335-387 (2008). Let \(\Gamma\) be the fundamental group of a compact Riemannian manifold \(X\) of sectional curvature \(K\leq -1\) and dimension \(n\geq 3\). We suppose that \(\Gamma=A*_C B\) is the free product of its subgroups \(A\) and \(B\) amalgamated over the 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 \(Y\subset X\), totally geodesic, of constant sectional curvature \(-1\), whose fundamental group is \(C\) and which separates \(X\) in two connected components whose fundamental groups are \(A\) and \(B\), respectively. Similar results hold if \(\Gamma\) is an HNN extension, or, more generally, if \(\Gamma\) acts on a simplicial tree without fixed point. Cited in 1 ReviewCited in 8 Documents MSC: 53C20 Global Riemannian geometry, including pinching 57M05 Fundamental group, presentations, free differential calculus Keywords:fundamental group; critical exponent; totally geodesic; constant sectional curvature PDFBibTeX XMLCite \textit{G. Besson} et al., J. Differ. Geom. 79, No. 3, 335--387 (2008; Zbl 1206.53038) Full Text: DOI arXiv Euclid