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Register a new userRigidity of graph products of abelian groups
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Authors
Mauricio Gutierrez
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We show that if $G$ is a group and $G$ has a graph-product decomposition with finitely-generated abelian vertex groups, then $G$ has two canonical decompositions as a graph product of groups: a unique decomposition in which each vertex group is a directly-indecomposable cyclic group, and a unique decomposition in which each vertex group is a finitely-generated abelian group and the graph satisfies the $T_0$ property. Our results build on results by Droms, Laurence and Radcliffe.
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For a group $G,$ let $Gamma(G)$ denote the graph defined on the elements of $G$ in such a way that two distinct vertices are connected by an edge if and only if they generate $G$. Moreover let $Gamma^*(G)$ be the subgraph of $Gamma(G)$ that is induced by all the vertices of $Gamma(G)$ that are not isolated. We prove that if $G$ is a 2-generated non-cyclic abelian group then $Gamma^*(G)$ is connected. Moreover $mathrm{diam}(Gamma^*(G))=2$ if the torsion subgroup of $G$ is non-trivial and $mathrm{diam}(Gamma^*(G))=infty$ otherwise. If $F$ is the free group of rank 2, then $Gamma^*(F)$ is connected and we deduce from $mathrm{diam}(Gamma^*(mathbb{Z}times mathbb{Z}))=infty$ that $mathrm{diam}(Gamma^*(F))=infty.$
Given a finite simplicial graph $Gamma=(V,E)$ with a vertex-labelling $varphi:Vrightarrowleft{text{non-trivial finitely generated groups}right}$, the graph product $G_Gamma$ is the free product of the vertex groups $varphi(v)$ with added relations that imply elements of adjacent vertex groups commute. For a quasi-isometric invariant $mathcal{P}$, we are interested in understanding under which combinatorial conditions on the graph $Gamma$ the graph product $G_Gamma$ has property $mathcal{P}$. In this article our emphasis is on number of ends of a graph product $G_Gamma$. In particular, we obtain a complete characterization of number of ends of a graph product of finitely generated groups.
We obtain a complete classification of graph products of finite abelian groups whose Cayley graphs with respect to the standard presentations are planar.
We study coherence of graph products and Coxeter groups and obtain many results in this direction.
We show that the automorphism group of a graph product of finite groups $Aut(G_Gamma)$ has Kazhdans property (T) if and only if $Gamma$ is a complete graph.
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