On the fundamental group of some open manifolds (Q2641697)

The author studies fundamental groups of non-compact Riemannian manifolds. Whereas the first theorem assumes a lower sectional curvature bound and states that, under this assumption, one obtains a lower bound for the length of the shortest homotopically nontrivial geodesic loop, provided the volume of large distance balls is sufficiently bounded, the following theorems drop the curvature condition and replace it by appropriate geometric conditions, in order to obtain related topological statements. The main theorems are precisely: Theorem 1.1. Let \(M^n\) be a complete \(n\)-dimensional noncompact Riemannian manifold, and fix a point \(p\in M\). Assume that the sectional curvature \(K\) is bounded below, \(K\geq 1\). For any real number \(L>0\), there exists a constant \(v=v(L,n)\in (0,1)\) such that, if \(\lim_{r\rightarrow \infty} \frac{\operatorname {vol} B(p,r)}{\text{vol}_{\mathbb{H}^n}(r)}\geq 1-v\) and \(M\) is not simply connected, then the length of the shortest homotopically nontrivial geodesic loop based at \(p\) is bigger than \(L\). Theorem 1.3. Let \(\widetilde{M}\) be a complete noncompact Riemannian manifold and let \(\widetilde{p}\in \widetilde{M}\) be a point. Let \(G\) be a discrete group of isometries acting freely on \(\widetilde{M}\). If \(\widetilde{M}\) has small diameter growth \(\limsup_{r\rightarrow \infty} \frac{\operatorname {diam}\partial B(\widetilde{p},r)}{r}<1\), then either \(\widetilde{M}/G\) is compact or \(G\) is finite. Theorem 1.6. Let \(M\) be a complete noncompact Riemannian manifold and let \(p\in M\) be a point. Consider the universal Riemannian cover \(\widetilde{M}\) and choose a lift \(\widetilde{p}\in \widetilde{M}\) of \(p\). If \(\widetilde{M}\) has the thick hinges property at \(\widetilde{p}\), the fundamental group \(\pi_1(M,p)\) is finitely generated. Here a Riemannian manifold \(M\) has the thick hinges property at \(p\), if there exists a constant \(\theta_0>0\) such that if \((p,\gamma_1,\gamma_2)\) is a minimizing geodesic hinge at \(p\) with sidelengths \(L\) and \(d(\gamma_1(l),\gamma_2(L))\geq L\), then the angle of the hinge at \(p\) is greater than or equal to \(\theta_0\).