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Nielsen zeta function, 3-manifolds, and asymptotic expansions in Nielsen theory. (English. English summary) Zbl 1023.55002

Let \(X\) be a connected compact polyhedron, \(f:X\to X\) a continuous map, and \(N(f)\) the Nielsen number of \(f\). The first main result of the paper asserts that if \(f\) is an orientation-preserving homomorphism of a special Haken or a special Seifert 3-manifold \(X\), then the Nielsen zeta function \[ N_f(z)=\exp\left(\sum_{n=1}^\infty \frac{N(f^n)}{n}z^n\right) \] [V. B. Pilyugina and A. L. Fel’shtyn, Funct. Anal. Appl. 19, 300-305 (1985; Zbl 0603.58041)] is either a rational function or a rational of a rational function. Let \(X\) be a closed surface of negative Euler characteristic, \(f:X\to X\) a pseudo-Anosov homeomorphism, \(\widetilde{f}:\widetilde{X}\to \widetilde{X}\) a fixed lifting of \(f\) to the universal covering \(\widetilde{X}\) of \(X\) and \(\widetilde{f}_*\) the endomorphism of the fundamental group \(\pi_1(X)\) defined by the equality \(\widetilde{f}_*(\gamma)\circ \widetilde{f}=\widetilde{f}\circ\gamma\) for all \(\gamma\in\pi_1(X)\). The second main result gives an asymptotic expansion for three equal numbers: the number of fixed point classes of \(f\) of norm less than \(x\), the number of lifting classes of \(f\) of norm less than \(x\), and the number of twisted conjugacy classes of norm less than \(x\) for \(\widetilde{f}_*\) in \(\pi_1(X)\), where the norm of a fixed point class or the corresponding twisted conjugacy class is defined as the length of the corresponding primitive closed geodesic in the mapping torus of \(f\). A generalization of this result to manifolds of higher dimension is also given.
Reviewer: E.S.Golod (Moskva)

MSC:

55M20 Fixed points and coincidences in algebraic topology
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
57M99 General low-dimensional topology
54H20 Topological dynamics (MSC2010)

Citations:

Zbl 0603.58041
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