Preservation and distortion of area in finitely presented groups
type: Item
instance of: scholarly article
title: Preservation and distortion of area in finitely presented groups
zbMATH Open document ID: 0868.20033
DOI: 10.1007/BF02247889
published in: Geometric and Functional Analysis. GAFA
publication date: 1997-06-18T00:00:00Z
full work available at URL: https://eudml.org/doc/58228
review text: The author develops the theory of distortion of combinatorial area in the Cayley graphs of finitely presented groups. Let \(G\) be a group with finite presentation \(\mathcal P\). The author defines an automorphism \(\phi\) of \(G\) to be \(\mathcal P\)-tame if \(\phi\) lifts to an automorphism of the free group on the generators of \(\mathcal P\) preserving the normal subgroup relations. He proves the following theorem: If \(\mathcal P\) is a finite one-relator presentation of \(G\) and if \(\phi\) is a \(\mathcal P\)-tame automorphism of \(G\), then there exists \(C>0\) so that for all loops \(w\) in the Cayley graph of \(\mathcal P\), we have \(\text{Area}_{\mathcal P}(w)\leq C \text{Area}_{\mathcal Q}(w)\), where \({\mathcal Q}\) is a finite presentation for the semi-direct product over \(\phi\) of \(G\) with the group \(\mathbb{Z}\) of integers. Here, the area \(\text{Area}_{\mathcal P}(w)\) of the loop \(w\) in the presentation \({\mathcal P}\) is the minimal number of 2-cells in a van Kampen diagram for \(w\). The author proves then that if \(K\) is the semi-direct product over \(\phi\) of the finitely presented group \(G\) with \(\mathbb{Z}\), then the area distortion of the subgroup \(G\) of \(K\) is at most an exponential of an isoperimetric function for \(K\). He presents examples of such a situation, where the inclusion \(G<K\) does not distort length but where area is distorted, and examples where length is distorted but area undistorted. The author introduces the exponent of area distortion \(\alpha(G,H)\) for the inclusion of a finitely presented group \(H\) in a finitely presented group \(G\), when \(H\) has a polynomial isoperimetric function of degree \(d\), and he proves that \(1\leq\alpha(G,H)\leq d\). Another result is the following suspension theorem: If \(\phi\in\text{GL}_n(\mathbb{Z})\) is of infinite order, then the exponent of area distortion of the torus-bundle over the circle with monodromy \(\phi\oplus 1\in\text{GL}_{n+1}(\mathbb{Z})\) i…
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20F06
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 57M07
zbMATH DE Number: 912455
zbMATH Keywords: distortion of combinatorial area
zbMATH Keywords: Cayley graphs of finitely presented groups
zbMATH Keywords: tame automorphisms
zbMATH Keywords: finite presentations
zbMATH Keywords: numbers of \(2\)-cells
zbMATH Keywords: van Kampen diagrams
zbMATH Keywords: semi-direct products
zbMATH Keywords: isoperimetric functions
zbMATH Keywords: exponent of area distortion
zbMATH Keywords: automatic groups
zbMATH Keywords: fundamental groups
author: Steve M. Gersten
reviewed by: Athanase Papadopoulos
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 912455