Higher-dimensional Reidemeister torsion invariants for cusped hyperbolic 3-manifolds (Q2879005)

Given a cusped hyperbolic \(3\)-manifold \(M\), the authors introduce and study a family of invariants \({\mathcal T}_n(M,\eta)\), depending on an integer \(n\geq 4\) and a choice of spin structure \(\eta\). The main ingredient to construct \({\mathcal T}_n(M,\eta)\) is a well-chosen representation \(\rho_n\) of the fundamental group of \(M\) into \(SL(n,{\mathbf C})\).NEWLINENEWLINESince \(M\) is hyperbolic, its fundamental group admits a discrete and faithful representation into \(PSL(2,{\mathbf C})\), \textsl{the holonomy}, which is unique up to conjugacy. The holonomy representation always lifts to a representation in \(SL(2,{\mathbf C})\) and such lifts are parametrised by the set of spin structure on \(M\). It is well-known that for \(n\geq3\), there is a unique irreducible representation \(\zeta_n\) of \(SL(2,{\mathbf C})\) into \(SL(n,{\mathbf C})\) given by the action of \(SL(2,{\mathbf C})\) on the \(n\)th symmetric symmetric power of \({\mathbf C}^2\). The representation \(\rho_n\), which depends on \(\eta\), is then obtained by lifting the holonomy to \(SL(2,{\mathbf C})\) according to \(\eta\) and then composing with \(\zeta_n\).NEWLINENEWLINEGiven the representation \(\rho_n\) one might want to compute its associated Reidemeister torsion. In this case, however, since \(M\) is not closed, the cohomology with local coefficients \(H^*(M;\rho_n)\) does not vanish and the torsion \(\tau(M;\rho_n;\{\theta_i\})\) depends on a choice of basis \(\{\theta_i\}\). The authors prove that one can choose \(\{\theta_i\}\) in such a way that the quotients NEWLINE\[NEWLINE{\mathcal T}_n(M,\eta)={{\tau(M;\rho_n;\{\theta_i\}}}\over{\tau(M;\rho_s;\{\theta_i\})}NEWLINE\]NEWLINE where \(s=2\) if \(n\) is even and \(s=3\) if \(n\) is odd, do not depend on \(\{\theta_i\}\).NEWLINENEWLINEThe study of these invariants leads the authors to prove two main results. The first result concerns the asymptotic behaviour of the invariants: the authors show that for a well-chosen spin structure one has NEWLINE\[NEWLINE\lim_{n\rightarrow \infty}{{\log |{\mathcal T}_n(M,\eta)|\over{n^2}}}=-{{\text{Vol}(M)}\over{4\pi}}NEWLINE\]NEWLINE generalising a result obtained by Müller in the closed case. In the cusped case some of the tools exploited by Müller (like Ray-Singer torsion) are no longer available. The authors overcome such difficulties by approximating \(M\) by compact manifolds \(M{p/q}\) obtained by Dehn surgery on \(M\) and relating the Reidemeister torsion of \(M\) and \(M{p/q}\).NEWLINENEWLINEFor odd \(n\), the invariant \({\mathcal T}_n(M,\eta)\) does not depend on the choice of spin structure, so the above result can be restated in the following way: NEWLINE\[NEWLINE\lim_{k\rightarrow \infty}{{\log |{\mathcal T}_{2k+1}(M)|\over{(2k+1)^2}}}=-{{\text{Vol}(M)}\over{4\pi}}NEWLINE\]NEWLINENEWLINENEWLINEThe second result concerns the relation between the invariants and the complex-length spectrum of the manifold: the authors show that for all \(N\geq 4\) the sequence \(\{|{\mathcal T}_{2k+1}(M)|\}_{k\geq N}\) determines the complex-length spectrum of \(M\) up to complex conjugation.NEWLINENEWLINETo prove the above, the authors also show the following interesting fact: the map defined from the set of complete hyperbolic \(3\)-manifolds with finite volume endowed with the geometric topology to the set of measures on the exterior of the closed unit disc in \({\mathbf C}\) endowed with the topology of weak convergence that assigns to each manifold its complex length spectrum is continuous.