The fully residually \(F\) quotients of \(F*\langle x,y\rangle\).
type: Item
instance of: scholarly article
title: The fully residually \(F\) quotients of \(F*\langle x,y\rangle\).
zbMATH Open document ID: 1254.20024
DOI: 10.4171/GGD/154
published in: Groups, Geometry, and Dynamics
publication date: 2012-03-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/0810.1509
review text: The author describes the fully residually \(F\) groups, or limit groups relative to a free group \(F\), that are quotients of \(F*\langle x,y\rangle\). He shows the following: Theorem 1.1. Let \(F_{R(S)}\), a fully residually \(F\) quotient of \(F*\langle x,y\rangle\), be freely indecomposable modulo \(F\). The underlying graph \(X\) of its cyclic JSJ decomposition modulo \(F\) has at most 3 vertices, and \(X\) has at most two cycles. All vertex groups except maybe the vertex group \(F\leq\widetilde F\leq F_{R(S)}\) are either free of rank 2 or free Abelian. In all cases the vertex group \(\widetilde F\) can be generated by \(F\) and two other elements. Finally \(F_{R(S)}\) has uniform hierarchical depth relative to \(F\) at most 4.
reviewed by: Stephan Rosebrock
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20E26
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20F70
Mathematics Subject Classification ID: 20E08
zbMATH DE Number: 6019683
zbMATH Keywords: limit groups
zbMATH Keywords: equations over free groups
zbMATH Keywords: free products
zbMATH Keywords: JSJ decompositions
zbMATH Keywords: graphs of groups
zbMATH Keywords: freely indecomposable groups
author: Nicholas W. M. Touikan
MaRDI profile type: Publication
arXiv ID: 0810.1509
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OpenAlex ID: W1760478198
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