Dehn functions and finiteness properties of graph products
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.
scientific article; zbMATH DE number 896029