Endomorphisms of hyperbolic groups. I: The Hopf property
type: Item
instance of: scholarly article
title: Endomorphisms of hyperbolic groups. I: The Hopf property
zbMATH Open document ID: 0929.20033
DOI: 10.1016/S0040-9383(98)00015-9
published in: Topology
publication date: 1999-11-08T00:00:00Z
review text: A group is said to have the Hopf property if every surjective endomorphism of this group is injective. The author proves that a torsion-free hyperbolic group has the Hopf property. The main tool used in the proof is the canonical (JSJ) decomposition of a hyperbolic group which the author has developed in a series of previous papers (in part with E. Rips), where he established in this abstract group-theoretic setting the theory of Jaco-Shalen and Johannson of the canonical decomposition of a 3-manifold by using the characteristic submanifold. As a corollary, the author proves that if \(M\) and \(N\) are closed negatively curved manifolds and if there exist degree 1 maps from \(M\) to \(N\) and from \(N\) to \(M\), then \(M\) is homotopy equivalent to \(N\). It is known then that this implies, if \(\dim(M)\geq 5\), that \(M\) and \(N\) are homeomorphic.
Mathematics Subject Classification ID: 20F67
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 20E36
zbMATH DE Number: 1307828
zbMATH Keywords: finitely presented groups
zbMATH Keywords: hyperbolic groups
zbMATH Keywords: characteristic submanifolds
zbMATH Keywords: Hopf property
zbMATH Keywords: JSJ decomposition
zbMATH Keywords: endomorphisms
reviewed by: Athanase Papadopoulos
MaRDI profile type: Publication
author: Zlil Sela
description: scientific article; zbMATH DE number 1307828