Rigidity of amalgamated products in negative curvature

Abstract

Let Γ be the fundamental group of a compact riemannian manifold X of sectional curvature K ≤ −1 and dimension n ≥ 3. We suppose that Γ = A * C B is the free product of its subgroups A and B over the amalgamated subgroup C. We prove that the critical exponent δ(C) of C satisfies δ(C) ≥ n − 2. The equality happens if and only if there exist an embedded compact hypersurface Y ⊂ 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. Similar results hold if Γ is an HNN extension, or more generally if Γ acts on a simplicial tree without fixed point.Date: November 16, 2018.

Homological Dimension and Critical Exponent of Kleinian Groups

“…The proofs of our results are generalizations of the proofs due to Besson, Courtois and Gallot in [BCG2]. Our main contribution in comparison to their paper is treatment of arbitrary coefficient modules, working with relative homology groups and handling manifolds whose injectivity radius is not bounded from below.…”

“…We then extend ζ to the ǫ-thin part of M , to a locally finite absolute cycleζ of finite volume. Besson, Courtois and Gallot in [BCG2] proved existence of a natural map F : M → M which is (properly) homotopic to the identity and satisfies…”

“…Our main contribution in comparison to their paper is treatment of arbitrary coefficient modules, working with relative homology groups and handling manifolds whose injectivity radius is not bounded from below. The most nontrivial technical ingredient of our paper is the existence of the natural maps introduced in [BCG2] and their properties established in that paper.…”

“…Most of this paper was written when the author was visiting the Max Plank Institute for Mathematics in Bonn. I am grateful to Gérard Besson and Gilles Courtois for sharing with me an early version of [BCG2], to Leonid Potyagailo for motivating discussions and to…”

Volume comparison without curvature-assumption

“…For every ε > 0 such that ε < 1 7 3 (n+1) 3 1 κ min κ · inj(X, g 0 ) , 1 3n 3 3 , if there exists a continuous Gromov-Hausdorff ε-approximation h : (M, g) → (X, g 0 ) of non-zero absolute degree, 1 then Vol(M, g) > 1 + 18 (n + 2) 2 (κ ε) 1/3 −n/2 Vol(X, g 0 ) . (1.1) In this Theorem, we point out the fact that the ε-quasi inverse of h is not supposed continuous, and that it is impossible to get rid of the assumption of non-triviality of the absolute degree, as proved by the counter-examples of Sect.…”