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Register a new userOn coherence of graph products and Coxeter groups
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Authors
Olga Varghese
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We study coherence of graph products and Coxeter groups and obtain many results in this direction.
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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 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.
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.
We obtain a complete classification of graph products of finite abelian groups whose Cayley graphs with respect to the standard presentations are planar.
We use probabilistic methods to prove that many Coxeter groups are incoherent. In particular, this holds for Coxeter groups of uniform exponent > 2 with sufficiently many generators.
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