Rank gradient versus stable integral simplicial volume
If \(\Gamma\) is a finitely generated group and \(\Gamma_*=(\Gamma_k)_{k\in\mathbb N}\) is a chain of finite index subgroups of \(\Gamma\), then the rank gradient of \(\Gamma\) with respect to \(\Gamma_*\) is defined as \(\operatorname{rg}(\Gamma,\Gamma_*)=\inf\limits_{k\in\mathbb N}\frac{d(\Gamma_k)-1}{[\Gamma:\Gamma_k]}\), while the absolute rank gradient of \(\Gamma\) is defined as \(\operatorname{rg}(\Gamma)=\inf\limits_{H\in F(\Gamma)}\frac{d(H)-1}{[\Gamma:H]}\), where \(F(\Gamma)\) is the set of all finite index subgroups of \(\Gamma\) and \(d(\Gamma)\) is the rank of \(\Gamma\), i.e., the minimal size of a generating set of \(\Gamma\). If \(\Gamma\) is the fundamental group of an oriented closed connected \(n\)-manifold \(M\), \(\Gamma_*=(\Gamma_k)_{k\in\mathbb N}\) is a chain of finite index subgroups of \(\Gamma\), and \(M_k=\widetilde M/\Gamma_k\) is the covering manifold of \(M\) associated with the subgroup \(\Gamma_k\), then the stable integral simplicial volume of \(M\) with respect to \(\Gamma_*\) is defined as \(\| M\| ^{\Gamma_*}_{\mathbb Z}=\inf\limits_{k\in\mathbb N}\frac{\| M_k\|_{\mathbb Z}}{[\Gamma:\Gamma_k]}\), while the stable integral simplicial volume of \(M\) is defined as \(\| M\| ^{\infty}_{\mathbb Z}=\inf\limits_{H\in F(\Gamma)}\frac{\| \widetilde M\|_{\mathbb Z}}{[\Gamma:H]}\). In this paper, the author observes that stable integral simplicial volume of closed manifolds gives an upper bound for the rank gradient of the corresponding fundamental groups. It is shown that if \(M\) is an oriented closed connected manifold with fundamental group \(\Gamma\) and \(\Gamma_*=(\Gamma_k)_{k\in\mathbb N}\) is a chain of finite index subgroups of \(\Gamma\), then \(\operatorname{rg}(\Gamma,\Gamma_*)\leq \| M\| ^{\Gamma_*}_{\mathbb Z}\). In particular, \(\operatorname{rg}(\Gamma)\leq \| M\| ^{\infty}\).
scientific article; zbMATH DE number 6865420
scientific article