Polyhedral groups and graph amalgamation products

Ordering Graph Products of Groups

Let Γ be a finite simple graph with vertex set V , so the edges may be taken to be unordered pairs of distinct elements of V . Assume that, for every v ∈ V , there is assigned a group Gv. The graph product GΓ is the quotient of the free product ∗v∈V Gv by the normal subgroup generated by all [Gu,Gv] for which {u,v} is an edge. (As usual, [Gu,Gv] means the subgroup generated by { x−1y−1xy | x ∈ ...

Free Products with Amalgamation and 3-manifolds

Rigidity of Graph Products of Abelian Groups

We show that if G is a group and G has a graphproduct 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...

The Topology of Graph Products of Groups

Given a finite (connected) simplicial graph with groups assigned to the vertices, the graph product of the vertex groups is the free product modulo the relation that adjacent groups commute. The graph product of finitely presented infinite groups is both semistable at infinity and quasi-simply filtrated. Explicit bounds for the isoperimetric inequality and isodiametric inequality for graph prod...

Random graph products of finite groups are rational duality groups

Given a binomial random graph G(n, p), we determine various facts about the cohomology of graph products of groups for the graph G(n, p). In particular, the random graph product of a sequence of finite groups is a rational duality group with probability tending to 1 as n → ∞. This includes random right angled Coxeter groups as a special case. AMS classification numbers. Primary: 05C10, 05C80, 2...