Automorphisms of negatively curved polygonal amalgams
type: Item
instance of: scholarly article
title: Automorphisms of negatively curved polygonal amalgams
zbMATH Open document ID: 0818.20022
author: John Meier
published in: Michigan Mathematical Journal
publication date: 1995-08-16T00:00:00Z
review text: Polygonal amalgams are the 2-dimensional analogues of free products with amalgamation. More precisely, a polygon of groups is a contravariant functor from the partially ordered set of the faces of a polygon (of dimensions 0, 1 and 2), ordered by inclusion, to the category of groups and monomorphisms; a polygonal amalgam is the direct limit of such a diagram of groups. A polygonal amalgam is called developable if it comes from an action of the group on a 1-connected 2-complex, transitively on the 2-cells (then the action has a single 2-cell as fundamental domain, and the original polygon of groups is recovered by associating to the faces of this 2-cell their stabilizers under the action). For example, a ``negatively curved'' polygonal amalgam is developable, in particular to vertex groups inject into the group; this result is due to Gersten and Stallings who defined the notion of a negatively curved amalgam by means of a suitable ``angle condition'' for the polygon of groups. ``By studying the actions of negatively curved polygonal amalgams on their corresponding 1-connected 2-complexes, we are able to describe the automorphism group of the amalgams. In particular we show that the automorphism group of a negatively curved polygonal amalgam of finite groups is virtually again a negatively curved polygonal amalgam of finite groups. Also, negatively curved polygonal amalgams of finite groups are co-Hopfian.'' The main auxiliary result is a rigidity property which says that, under some hypothesis satisfied for example for amalgams with finite vertex groups, any automorphism of a negatively curved amalgam takes the stabilizers of the faces of a 2-cell (chamber) of the associated 2- complex again to the stabilizers of the faces of some 2-cell.
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20F34
Mathematics Subject Classification ID: 20E36
zbMATH DE Number: 575625
zbMATH Keywords: developable polygonal amalgams
zbMATH Keywords: free products with amalgamation
zbMATH Keywords: polygon of groups
zbMATH Keywords: diagram of groups
zbMATH Keywords: actions
zbMATH Keywords: automorphism groups
zbMATH Keywords: negatively curved polygonal amalgam of finite groups
reviewed by: Bruno Zimmermann
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029004919
OpenAlex ID: W2005868725
DOI: 10.1307/MMJ/1029004919
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description: scientific article; zbMATH DE number 575625