Higher dimensional Reidemeister torsion invariants for cusped hyperbolic 3-manifolds

Details Higher dimensional Reidemeister torsion invariants for cusped hyperbolic 3-manifolds Higher dimensional Reidemeister torsion invariants for cusped hyperbolic 3-manifolds

2011-10-17

arxiv.org/abs/1110.3718v1

Mathematics

Geometric Topology

Scientific paper

For an oriented finite volume hyperbolic 3-manifold $M$ with a fixed spin structure $\eta$, we consider a sequence of invariants $\{\normaltor_n(M; \eta)\}$. Roughly speaking, $\normaltor_n(M; \eta)$ is the Reidemeister torsion of $M$ respect to the representation given by the composition of the lift of the holonomy representation defined by $\eta$, and the $n$-th dimensional irreducible complex representation of $\sln(2,\cmplx)$. 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, $\log |\normaltor_n(M; \eta)| \sim -n^2 \frac{\Vol M}{4\pi}$, extending thus the result obtained by W.\;M\"uller for the compact case in \cite{Mul}. Concerning the latter, we prove that the sequence $\{|\normaltor_n(M; \eta)|\}$ determines the complex length spectrum of the manifold up to complex conjugation.

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