Dehn functions and finiteness properties of graph products
type: Item
instance of: scholarly article
title: Dehn functions and finiteness properties of graph products
zbMATH Open document ID: 0861.20027
DOI: 10.1016/0022-4049(95)00028-3
author: Yanyan Li
published in: Journal of Pure and Applied Algebra
publication date: 1996-09-01T00:00:00Z
review text: Let \(\Gamma\) be a finite simplicial graph with vertices \(v_1,\dots,v_m\). Suppose that to each vertex \(v_i\) we have assigned a nontrivial group \(G_i\). Then the graph product \(G\Gamma\) is defined to be the quotient of the free product \(\coprod_{1\leq i\leq m}G_i\) by the normal subgroup generated by all \([G_i,G_j]\) for which \(\{v_i,v_j\}\) is an edge of \(\Gamma\). If \(f,g:\mathbb{N}\setminus\{0\}\to(0,\infty)\) are functions then \(f\preccurlyeq g\) if there are non-negative constants \(A\), \(B\), \(C\) such that \(f(n)\leq Ag(Bn)+Cn\). We say that \(f\), \(g\) are of the same type (\(f\equiv g\)) if \(f\preccurlyeq g\) and \(g\preccurlyeq f\). The class of \(f\) is denoted by \([f]\). Given a finitely presented group \(G\) the Dehn function of the group, \(\delta_G\), is defined to be \(\delta_G=[\delta P]\) where \(\delta P\) is the Dehn function of some finite presentation \(P\) of \(G\). The main results of this paper are Theorem 1. Let \(G\Gamma\) be a graph product. Then (i) if all the vertex groups are polynomial and for some \(i\), \(\text{deg }(G_i)\geq 2\), then \(G\Gamma\) is polynomial and \(\text{deg }(G\Gamma)=\max\{\text{deg}(G_i)\mid 1\leq i\leq m\}\). (ii) If for all \(i\) \(\text{deg}(G_i)=1\), then \(1\preccurlyeq\delta_{\text{GL}} \preccurlyeq 2\). (iii) If \(\Gamma\) contains an edge \(\{v_i,v_j\}\) with \(G_i\), \(G_j\) polynomial infinite groups then \(2\preccurlyeq \delta_{G\Gamma}\). (iv) \(\delta_{G\Gamma}\) is exponential if and only if \(\max\{\delta_{G_i}\mid 1\leq i\leq m\}\) is exponential. Theorem 2. Let \(1\leq n\leq \infty\). Then the graph product \(G\Gamma\) is \(FP_n\), \(F_n\), of finite cohomological dimension, or \(FP\), if and only if the same is true for each vertex group. Moreover, \(G\Gamma\) is geometrically finite if each vertex group is geometrically finite.
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20J05
zbMATH DE Number: 896029
zbMATH Keywords: geometrically finite groups
zbMATH Keywords: finite simplicial graphs
zbMATH Keywords: graph products
zbMATH Keywords: free products
zbMATH Keywords: finitely presented groups
zbMATH Keywords: Dehn functions
zbMATH Keywords: finite presentations
reviewed by: L. A. Kurdachenko
MaRDI profile type: Publication
full work available at URL: 0022-4049(95)00028-3
OpenAlex ID: W1969354875
cites work: Q3350312
cites work: Q4028716
cites work: Finiteness conditions on groups and quasi-isometries
cites work: On Dehn Functions and Products of Groups
cites work: Q4143495
cites work: The Growth Series of a Graph Product
cites work: Isoperimetric functions for graph products
cites work: Projective resolutions for graph products
Wikidata QID: Q128109132
description: scientific article; zbMATH DE number 896029