Higher‐dimensional Reidemeister torsion invariants for cusped hyperbolic 3‐manifolds
Higher‐dimensional Reidemeister torsion invariants for cusped hyperbolic 3‐manifolds
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尖点双曲 3 流形的高维 Reidemeister 扭转不变量
DOI：
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发表时间：
2011
期刊：
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通讯作者：
J. Porti
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作者：
P. Menal;J. Porti
对于具有固定自旋结构η的定向有限体积双曲三维流形M，我们考虑一个不变量序列Tn（M; η）}。粗略地说，Tn（M; η）是M相对于由η定义的完整表示的提升和SL（2，C）的n维不可约复表示的合成给出的表示的雷德迈斯特挠率。在目前的工作中，我们专注于这个不变量的两个方面：它的渐近行为和它与流形的复长谱的关系。关于前者，我们证明，对于适当的自旋结构，log| Tn（M; η）|−n2（关于后者，我们证明了序列{|Tn（M; η）|}确定流形的复长谱直到复共轭。
For an oriented finite volume hyperbolic 3‐manifold M with a fixed spin structure η, we consider a sequence of invariants Tn(M; η)}. Roughly speaking, Tn(M; η) is the Reidemeister torsion of M with respect to the representation given by the composition of the lift of the holonomy representation defined by η, and the n‐dimensional, irreducible, complex representation of SL(2, C). In the present work, we focus on two aspects of this invariant: its asymptotic behaviour and its relationship with the complex‐length spectrum of the manifold. Concerning the former, we prove that, for suitable spin structures, log | Tn(M; η)| ∼ −n2 (Vol M/4π), extending thus the result obtained by Müller for the compact case. Concerning the latter, we prove that the sequence {| Tn(M; η)|} determines the complex‐length spectrum of the manifold up to complex conjugation.