The theory of isometric group actions on R-trees is extended to actions by homeomorphisms with the following non-nesting property: no group element maps an arc properly into itself. A finitely presented group acting freely by homeomorphisms on an R-tree is free abelian or splits over a (possibly trivial) cyclic group. 1991 Mathematics Subject Classification 20E08, 20F32, 57M60.
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Gilbert Levitt (1998) studied this question.