Convergence of normalized Betti numbers in nonpositive curvature (Q6046450)

The authors study convergence of volume-normalized Betti numbers in Benjamini-Schramm convergent sequences of finite volume Riemannian manifolds. Let \(\mathscr{M}=\{\text{pointed Riemannian manifolds }(M,p)\}/\text{pointed isometry},\) endowed with the topology of pointed smooth convergence. For a connected, complete, finite-volume Riemannian manifold \(M\), there is a map from \(M\) to \(\mathscr{M}\) defined by \(p\to [(M,p)]\), and the Riemannian measure on \(M\) pushes forward to a measure \(\mu_{M}\) on \(\mathscr{M}\). We say a sequence \((M_n)\) Benjamini-Schramm converges if the measures \(\mu_{M_n}/\text{vol}(M_n)\) weakly converge to a probability measure on \(\mathscr{M}\). Moreover, if \(X\) is an irreducible symmetric space of noncompact type, then we say a sequence \((M_n)\) Benjamini-Schramm converges to \(X\) if \(\mu_{M_n}/\text{vol}(M_n)\) weakly converges to the atomic probability measure on \([(X,x)]\in \mathscr{M}\). The paper studies the following question: if \((M_n)\) Benjamini-Schramm converges (maybe to some \(X\)), do the volume-nomalized Betti numbers \((b_k(M_n)/\text{vol}(M_n))\) converge? One result is Theorem 1.2, which states that if \((M_n)\) is a Benjamini-Schramm convergent sequence of compact, \(\epsilon\)-thick Riemannian manifolds with upper and lower curvature bounds, then the volume normalized Betti numbers of \((M_n)\) converge. This result was first written up by \textit{L. Bowen} [Duke Math. J. 164, No. 3, 569--615 (2015; Zbl 1312.57041)], but Bowen's proof was not complete (as pointed out by the authors) and the authors give a slightly different proof here. Another result is Theorem 1.3, which states that if \((M_n)\) is a Benjamini-Schramm convergent sequence of finite-volume \(X\)-manifolds with \(\dim(X)\ne 3\), then the volume normalized Betti numbers of \((M_n)\) converge. The proof of Theorem 1.3 is divided into the following two cases. \begin{itemize} \item A sequence of Riemannian \(d\)-manifolds with sectional curvatures lying in \([-1,\delta]\), with \(-1\leq \delta<0\) and \(d\ne 3\). This case is proved in Theorem 1.6. \item A sequence of real-analytic Riemannian \(d\)-manifolds with sectional curvatures lying in \([-1,0]\), such that the univeral covers of \(M_n\) have no Euclidean deRham factors, and \((M_n)\) converges to a measure \(\mu\) on \(\mathscr{M}\) supported on \(\epsilon\)-thick manifolds. This case is proved in Theorem 1.7. \end{itemize} Note that both Theorem 1.6 and Theorem 1.7 work for general Riemannian manifolds, instead only for \(X\)-manifolds as in Theorem 1.3. Theorems 1.2, 1.6 and 1.7 are all applications of a more technical result: Theorem 2.9. Under some assumptions on upper and lower bounds of volumes of balls in \(M_n\) and an extra assumption, Theorem 2.9 proves that volume-normalized Betti numbers of some nerves of \(M_n\) converge. Theorem 2.9 is a (modified) weaker version of Theorem 4.1 of Bowen's work [loc. cit.], but the proof of Theorem 4.1 in Bowen's paper is not complete (as pointed out by the authors). The proof of Theorem 1.6 is a direct application of Theorem 2.9 to \((M_n)\), although checking the assumption of Theorem 2.9 is not a trivial work. Along the way, one important geometric estimate is Proposition 3.1, which states that when \(d\geq 4\), the number of Margulis tubes around very short geodesics is small compared to the volume of the manifold. To prove Theorem 1.7, the authors apply Theorem 2.9 to the so-called stable thick parts of \(M_n\). Then they take care of the stable thin part of \(M_n\) by proving that the Betti numbers of these thin parts and their boundaries are much smaller than the volume of \(M_n\). Theorem 1.3 does not work for the hyperbolic case (\(X=\mathbb{H}^3\)) since one can construct a sequence of hyperbolic \(3\)-manifolds \((M_n)\) by taking finite covers of Dehn fillings of a fixed cusped hyperbolic \(3\)-manifold. Moreover, this theorem holds for \(d\geq 4\) because the Proposition 3.1 mentioned above only holds when \(d\geq 4\).