An asymptotic formula for the eta invariants of hyperbolic 3-manifolds (Q2368010)

For a hyperbolic 3-manifold \(N\), the eta invariant \(\eta(N)\) measures the difference between the signature of a 4-manifold that \(N\) bounds and the integral of 1/3 of the first Pontryagin form of the 4-manifold. In case \(N\) is a manifold \(M(p,q)\) obtained by a \((p,q)\) Dehn filling of a hyperbolic \(3\)-manifold \(M\) with one cusp, \(\eta(M(p,q))\) is related to the volume of \(M(p,q)\) by a formula proven by \textit{T. Yoshida} [Invent. Math. 81, 473-514 (1985; Zbl 0594.58012)]. This formula involves the volume, the eta invariant, the complex length of the core geodesic of the filled-in torus, and an analytic function \(f\) defined on the Dehn surgery space. For the special case of Dehn fillings of the figure-8 knot complement, he gave a simpler version of the formula. In this paper, the authors show that the simpler version can be derived in general from Yoshida's result. More precisely, they show that for \(p^ 2 + q^ 2\) sufficiently large, the volume and eta invariant (times certain constants) are the real and imaginary parts of a number which is the sum of \(f(u(p,q))\), where \(f\) is an analytic function of the Dehn surgery parameter \(u\), plus simple terms involving the complex length and a constant \(l(p,q)\) related to the Hirzebruch defect \(\text{def}(p;q,1)\); \(l(p,q)\) can also be calculated easily from a continued fraction expansion of \(p/q\). The authors believe that the formula holds without the restriction on \(p^ 2 + q^ 2\). The proof of their formula uses a cobordism from \(M(p,q)\) to the union of a fixed Riemannian manifold \(M'\) and the lens space \(L(p,q)\). A result of C. T. C. Wall implies that the cobordism has signature \(0\), and a formula of the first author applies to give an expression for the eta invariant. The specific calculation of Yoshide in the figure-8 case identifies the unknown terms in this expression, giving the main result. A generalization to fillings of manifolds with several cusps is given. As an application, the authors use their formula and some clever calculations with the Hirzebruch defect to deduce that the set of eta invariants of \(M(p,q)\), for any \(M\) with one cusp, is dense in the real numbers. In a closing section, the cases of the figure-8 knot and the Whitehead link are worked out explicitly to illustrate the theory.