Pinching estimates for negatively curved manifolds with nilpotent fundamental groups (Q2499529)

The authors relate the nilpotency degree of the fundamental group to the pinching of the negatively curved manifold. First it is shown that for a complete Riemannian manifold \(M\) with sectional curvature satisfying \(-a^2\leq\text{sec} (M)\leq-1\), if \(\Gamma\) is a \(k\)-step nilpotent subgroup of the fundamental group \(\pi_1(M)\), then \(a>k\), if \(a\in[1,2)\), then \(\Gamma\) is abelian. Further, they prove the following theorem: Let \(M\) be a pinched negatively curved manifold whose fundamental group \(\pi_1(M)\) has a \(k\)-step nilpotent subgroup of finite index. Then \(M\) admits a complete Riemannian metric such that \(\text{sec}(M)\in[-(k+\varepsilon)^2,-1]\) for \(\varepsilon> 0\). Now, given a manifold \(M\) one defines \(\text{pinch}^{\text{diff}}(M)\) to be the infimum of all \(a^2\geq 1\) such that \(M\) admits a Riemannian metric such that \(-a^2\leq \text{sec}(M)\leq-1\), and \(\text{pinch}^{\text{top}}(M)\) to be the infimum of all \(\text{pinch}^{\text{diff}}(N)\) where \(N\) is homeomorphic to \(M\), and also \(\text{pinch}^{\text{hom}} (M)\) to be the infimum of \(\text{pinch}^{\text{diff}}(N)\) where \(N\) is manifold with \(\dim(N)=\dim(M)\) and is homotopy equivalent to \(M\). Then, combining these theorems it is proven that if \(M\) is a pinched negatively curved manifold such that \(\pi_1(M)\) has a \(k\)-step nilpotent subgroup of finite index, then \(\text{pinch}^{\text{diff}}(M)=\text{pinch}^{\text{top}}(M)=\text{pinch}(M)^{\text{hom}}=k^2.\)