Endomorphisms of mapping tori

TL;DR

This paper classifies certain mapping tori of Poincaré Duality groups using Hopf properties, extending known results for 3-manifolds and applying to hyperbolic group rigidity.

Contribution

It provides a new classification framework for mapping tori of Poincaré Duality groups based on Hopf properties, generalizing previous 3-manifold results.

Findings

Classification of mapping tori with non-zero Euler characteristic

Extension of rigidity results to these mapping tori

New proof of classification for fibered 3-manifolds

Abstract

We classify in terms of Hopf-type properties mapping tori of residually finite Poincar\'e Duality groups with non-zero Euler characteristic. This generalises and gives a new proof of the analogous classification for fibered 3-manifolds. Various applications are given. In particular, we deduce that rigidity results for Gromov hyperbolic groups hold for the above mapping tori with trivial center.