On fibers and accessibility of groups acting on trees with inversions

Rasheed Mahmood Saleh Mahmood

Abstract


Throughout this paper the actions of groups on graphs with inversions are
allowed. An element g of a group \(G\) is called inverter if there exists a tree
\(X\) where \(G\) acts such that \(g\) transfers an edge of \(X\) into its inverse.
\(A\) group \(G\) is called accessible if \(G\) is finitely generated and there
exists a tree on which \(G\) acts such that each edge group is finite, no vertex
is stabilized by $G$, and each vertex group has at most one end.

In this paper we show that if \(G\) is a group acting on a tree \(X\) such that if
for each vertex \(v\) of \(X\), the vertex group \(G_{v}\) of \(v\) acts on a tree
\(X_{v}\), the edge group \(G_{e}\) of each edge e of \(X\) is finite and contains
no inverter elements of the vertex group \(G_{t(e)}\) of the terminal \(t(e)\) of
$e$, then we obtain a new tree denoted \(\widetilde{X}\) and is called a fiber
tree such that \(G\) acts on \(\widetilde{X}\). As an application, we show that if
$G$ is a group acting on a tree \(X\) such that the edge group \(G_{e}\) for each
edge \(e\) of \(X\) is finite and contains no inverter elements of \(G_{t(e)}\), the
vertex \(G_{v}\) group of each vertex \(v\) of \(X\) is accessible, and the quotient
graph \(G\diagup X\) for the action of \(G\) on \(X\) is finite, then \(G\) is an
accessible group.


Keywords


Ends of groups, groups acting on trees, accessible groups

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References


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