Some consequences of Arthur's work on the spectrum and topology of hyperbolic varieties
The paper under review deals with spectral and topological aspects of hyperbolic manifolds. More precisely, let \(G\) be a semisimple \({\mathbb Q}\)-group not coming from a twisted form \(\text{}^3D_4\) or \(\text{}^6D_4\), and such that \(G({\mathbb R})\) is the product of \(\mathrm{SO}(n,1)\) with a compact group. The paper contains several striking results which are interesting on their own, we will mention a few. Let \(M\) be the \(n\)-dimensional manifold \(\Gamma\backslash{\mathbb H}^n\) where \({\mathbb H}^n=\mathrm{SO}(n,1)/\mathrm{SO}(n)\) is the real hyperbolic space and \(\Gamma\subset G\) is a torsion-free congruence discrete subgroup of \(G\). One of the main results in the paper under review can be stated as follows (Theorem 1.2). The \(L^2\)-spectrum of the Laplace operator on functions on \(M\) is contained in the set: \[ \bigcup_{0\leq j<\frac{n-1}{2}}\left\{\left(\tfrac{n-1}{2}\right)^2-\left(\tfrac{n-1}{2}-j\right)^2\right\}\bigcup\;\left[\left(\tfrac{n-1}{2}\right)^2 -\left(\tfrac{1}{2}-\tfrac{1}{N^2+1}\right)^2;+\infty\right[ \] where \(N=n\) if \(n\) is even, and \(N=n+1\) otherwise. This result is connected with a conjecture of \textit{M. Burger} et al. [Bull. Am. Math. Soc., New Ser. 26, No. 2, 253--257 (1992; Zbl 0762.22009)] predicting that the spectrum should be contained in \(\bigcup_{0\leq j<\frac{n-1}{2}}\big\{(\frac{n-1}{2})^2-(\frac{n-1}{2}-j)^2\big\}\bigcup\;[ (\frac{n-1}{2})^2;+\infty[\). As a byproduct, a result on the spectrum of the Laplacian on differential forms is proved. Actually, for \(0\leq k\leq\frac{n}{2}-1\), there exists a positive number \(\varepsilon(n,k)\) such that the first non-zero eigenvalue \(\lambda_1^k(M)\) of the Laplace operator on square integrable \(k\)-forms on \(M\) satisfies the following inequality: \[ \lambda_1^k\geq\varepsilon(n,k). \] Some results relating the (co-)homology of \(M\) with totally geodesic compact submanifolds are obtained. Let \(H\subset G\) be a semisimple group over \({\mathbb Q}\) which is stable under a Cartan involution of \(G\) such that \(H\) does not come from the twisted forms \({}^3D_4\) or \({}^6D_4\), and \(H({\mathbb R})\simeq \mathrm{SO}(k,1)\times(\text{compact})\) with \(k\leq n\). The totally geodesic immersion \(\Gamma\cap H\backslash{\mathbb H}^k\rightarrow\Gamma\backslash{\mathbb H}^n\) induces a map \[ H_\bullet(\Gamma\cap H\backslash{\mathbb H}^k)\rightarrow H_\bullet(\Gamma\backslash{\mathbb H}^n) \] between homology groups. Next, for \(g\in G({\mathbb Q})\), the immersion \((H\cap g^{-1}\Gamma g)\backslash{\mathbb H}^k\rightarrow\Gamma\backslash{\mathbb H}^n\) induces a map on cohomology groups \[ H^\bullet(\Gamma\backslash{\mathbb H}^n)\rightarrow \prod_{g\in G({\mathbb Q})}H^\bullet((H\cap g^{-1}\Gamma g)\backslash{\mathbb H}^k). \] Finally, one gets natural maps \[ H_\bullet(\mathrm{Sh}^0H)\rightarrow H_\bullet(\mathrm{Sh}^0G)\text{ and }H^\bullet(\mathrm{Sh}^0G)\rightarrow \prod_{g\in G({\mathbb Q})}H^\bullet(\mathrm{Sh}^0H) \] where \(H_\bullet(\mathrm{Sh}^0G)=\lim_{\leftarrow\;\Gamma\text{ cong.}}H_\bullet(\Gamma\backslash{\mathbb H}^n)\) and \(H^\bullet(\mathrm{Sh}^0G)=\lim_{\rightarrow\;\Gamma\text{ cong.}}H^\bullet(\Gamma\backslash{\mathbb H}^n)\); with similar definition for \(H_\bullet(\mathrm{Sh}^0H)\) and \(H^\bullet(\mathrm{Sh}^0H)\). Then Theorem 1.6 proved in the paper can be stated as follows. For any integer \(i\geq\frac{n}{2}\), the natural map \(H_i(\mathrm{Sh}^0H)\rightarrow H_i(\mathrm{Sh}^0G)\) is injective; while for any integer \(i\leq\frac{k}{2}\), the natural map \(H^\bullet(\mathrm{Sh}^0G)\rightarrow \prod_{g\in G({\mathbb Q})}H^\bullet(\mathrm{Sh}^0H)\) is injective.
scientific article; zbMATH DE number 6176662
scientific article