A splitting theorem for groups acting on quasi-trees
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Batty, Michael (2000). A splitting theorem for groups acting on quasi-trees. Communications in Algebra 28 (2), 967-980
It is well known that a group is free if and only if it acts freely without inversions on a tree. We prove a generalisation of this fact by defining a quasi-tree to be a graph with a bound on the size of its simple loops. It is shown that a finitely generated group acting freely on such a graph is isomorphic to a free product of free groups and finite groups.