Endomorphisms of mapping tori

Abstract: One of the most fundamental results in 3-dimensional topology, proved in works of Gromov, Mostow, Wang and Waldhausen, is that any self-map of non-zero degree of a mapping torus of a closed hyperbolic surface is homotopic to a homeomorphism if and only if the monodromy is not periodic. Key properties for the proof were the existence of hyperbolic structures or of non-vanishing semi-norms (such as the simplicial volume). Using Algebra, we give a new, unified proof and generalise the above result in every dimension, by replacing the hyperbolic surface with a corresponding higher dimensional aspherical manifold. More generally, we will classify in terms of Hopf-type properties mapping tori of residually finite Poincaré Duality groups with non-zero Euler characteristic. It turns out that the rigidity behavior of these mapping tori with trivial center is similar to that of non-elementary torsion-free hyperbolic groups.

algebraic topologydifferential geometrygeneral topologygroup theorygeometric topology

Audience: researchers in the topic

Ohio State Topology and Geometric Group Theory Seminar

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