Cheeger constants and \(L^2\)-Betti numbers. (English) Zbl 1312.57041

For a Riemannian manifold \(X\) an abstract group \(\Gamma\) satisfying \(\Gamma<\text{Isom}(X)\) is said to be geometric if its action is free and properly discontinuous. This ensures that \(X/\Gamma\) is a manifold and the quotient map \(X\to X/\Gamma\) is a cover. A residually finite countable group \(\Gamma\) has asymptotically vanishing lower \(d\) th Betti number if \(\liminf\frac{b_d(N)}{[\Gamma:N]}=0\), where the \(\liminf\) is with respect to the net of finite-index normal subgroups \(N\vartriangleleft\Gamma\) ordered by reverse inclusion, and \(b_d(N)\) is the \(d\)-dimensional \(L^2\)-Betti number of \(N\). The Cheeger constant of a smooth Riemannian manifold \(M\) is defined as \(h(M)=\inf\frac{\text{area}(\partial M_0)}{\text{vol}(M_0)}\), where the infimum is over all smooth compact submanifolds \(M_0\subset M\) with \(\text{vol}(M_0)\leq\text{vol}(M)/2\).
In this paper, the author proves the existence of positive lower bounds on the Cheeger constants of manifolds of the form \(X/\Gamma\), where \(X\) is a contractible Riemannian manifold and \(\Gamma<\text{Isom}(X)\) is a discrete subgroup. It is shown that if \(X\) is a smooth contractible complete Riemannian manifold, \(\mathcal{G}_d\) is the class of all residually finite countable groups \(\Gamma\) such that every finitely generated subgroup \(\Gamma'<\Gamma\) has asymptotically vanishing lower \(d\) th Betti number, and there is a residually finite geometric subgroup \(\Lambda<\text{Isom}(X)\) such that \(X/\Lambda\) is compact and \(b_d^{(2)}(\Lambda)>0\), then \(I(X|\mathcal{G}_d)=\inf_{\Gamma\in\mathcal{G}_d} h(X/\Gamma)>0\). As an application of this result, it is shown that if \(\Gamma\), where \(\Gamma<\text{Isom}(\mathbb H^n)\) and \(\mathbb H^n\) is a real hyperbolic \(n\)-space, is geometric and isomorphic with a subgroup of the fundamental group of a complete finite-volume hyperbolic \(3\)-manifold and \(n\geq 4\) is an even integer, then \(h(\mathbb H^n/\Gamma)\geq I(\mathbb H^n|\mathcal{G}_{n/2})>0\). In particular, \(I(\mathbb H^n|\text{Free})>0\) for every even integer \(n\geq 4\). This result implies the existence of a uniform positive upper bound on the Hausdorff dimension of the conical limit set of such a \(\Gamma\) when \(\Gamma\) is geometrically finite.


57S30 Discontinuous groups of transformations
22F30 Homogeneous spaces
Full Text: DOI arXiv Euclid


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