谱间隙、覆盖流形和剩余有限群的存在性
Existence of spectral gaps, covering manifolds and residually finite groups
摘要：
本文研究了具有剩余有限覆盖群$Γ$和紧基空间$(M,g)$的黎曼覆盖$(X,g) \to (M,g)$.特别是，我们给出了两个一般的程序，从而在一个家庭的变形覆盖$(X,g_\eps) \to (M,g_\eps)$，使频谱的拉普拉斯$Δ_{(X_\eps,g_\eps)}$至少有一个规定的有限数量的频谱间隙提供$\eps$是足够小的。如果$Γ$有一个正的Kadison常数，那么我们可以应用Brüning和Sunada的结果来推导出$\spec Δ_{(X,g_\eps)}$还有带结构，并且有一个关于$\spec {\laplacian {(X,g_\eps)}}$中与区间$[0,λ]$相交的分量个数$N(λ)$的渐近估计。我们还提出了几类剩余有限群的例子，适合我们的建设和研究它们之间的相互关系。最后，我们提到几个可能的应用程序，我们的结果。
英文摘要：
In the present paper we consider Riemannian coverings $(X,g) \to (M,g)$ with residually finite covering group $Γ$ and compact base space $(M,g)$. In particular, we give two general procedures resulting in a family of deformed coverings $(X,g_\eps) \to (M,g_\eps)$ such that the spectrum of the Laplacian $Δ_{(X_\eps,g_\eps)}$ has at least a prescribed finite number of spectral gaps provided $\eps$ is small enough. If $Γ$ has a positive Kadison constant, then we can apply results by Brüning and Sunada to deduce that $\spec Δ_{(X,g_\eps)}$ has, in addition, band-structure and there is an asymptotic estimate for the number $N(λ)$ of components of $\spec {\laplacian {(X,g_\eps)}}$ that intersect the interval $[0,λ]$. We also present several classes of examples of residually finite groups that fit with our construction and study their interrelations. Finally, we mention several possible applications for our results.
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