×

Rigidity of amalgamated products in negative curvature. (English) Zbl 1206.53038

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.

MSC:

53C20 Global Riemannian geometry, including pinching
57M05 Fundamental group, presentations, free differential calculus
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid