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CONTENTS

2.3 Eventually commutative endomorphisms 32

2.3.1 Trace formula for the Reidemeister numbers of eventu-

ally commutative endomorphisms 32

2.3.2 Rationality of Reidemeister zeta functions of eventu-

ally commutative endomorphisms - first proof 35

2.3.3 Functional equation for the Reidemeister zeta function

of an eventually commutative endomorphism 39

2.3.4 Rationality of Reidemeister zeta functions of eventu-

ally commutative endomorphisms - second proof. . . . 41

2.3.5 Connection of the Reidemeister zeta function with the

Lefschetz zeta function of the dual map 42

2.4 Endomorphisms of finite groups 44

2.5 Endomorphisms of the direct sum of a free Abelian and a

finite group 48

2.6 Endomorphisms of nilpotent groups 53

2.6.1 Functional equation 54

2.7 The Reidemeister zeta function and group extensions 55

2.8 The Reidemeister zeta function of a continuous map 56

2.8.1 The Reidemeister zeta function of a continuous map

and Serre bundles 59

3 The Nielsen zeta function 60

3.1 Radius of Convergence of the Nielsen zeta function 60

3.1.1 Topological entropy 60

3.1.2 Algebraic lower estimation for the Radius of Convergence 62

3.2 Nielsen zeta function of a periodic map 66

3.3 Orientation-preserving homeomorphisms of a compact surface 69

3.3.1 Geometry of the Mapping Torus and Radius of Con-

vergence 70

3.4 The Jiang subgroup and the Nielsen zeta function 72

3.5 Polyhedra with finite fundamental group 76

3.6 Nielsen zeta function in other special cases 82

3.6.1 Pseudo-Anosov homeomorphism of a compact surface 82

3.7 The Nielsen zeta function and Serre bundles 83

3.8 Examples 85