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Groups acting freely on \(\mathbb{R}\)-trees. (English) Zbl 0766.57021
The theory of actions of groups on simplicial trees, due to Bass and Serre, has numerous applications in topology and algebra, and its generalization to actions (by isometries) on \(\mathbb{R}\)-trees is also important. For example, the boundary points of certain spaces of representations of groups of interest in low-dimensional topology can be interpreted as actions on \(\mathbb{R}\)-trees obtained as limits of deformations of geometric structures. A longstanding and natural question in the theory of actions on \(\mathbb{R}\)-trees is to classify the groups which can act freely. Recently, E. Rips has proven the conjecture that the only such groups are free products of finitely generated free abelian groups and (torsionfree) fundamental groups of 2-manifolds. Rips’ result supersedes the main results of the paper under review, but the techniques and auxiliary results developed in the paper are nonetheless interesting and potentially useful.
The first main result is that a non-cyclic abelian subgroup of a finitely presented group acting freely on an \(\mathbb{R}\)-tree must be contained in an abelian subgroup which is a free factor. The proof uses measured laminations, and develops some technical results about neighborhoods of laminations in which some leaf is dense. The authors also prove an interesting statement in passing: when a group which is indecomposable with respect to free product with amalgamation or HNN decomposition (so the infinite cyclic group is not indecomposable in this sense) acts freely, then for every non-degenerate segment \(J\) the group is generated by the elements \(g\) such that \(g(J)\cap J\) is a nondegenerate segment.
The other main result is that if a finitely presented group acting freely on an \(\mathbb{R}\)-tree has an HNN decomposition \(H*_{\langle s\rangle}\) where \(\langle s\rangle\) is infinite cyclic, then there is a subgroup \(H'\) of \(H\) such that the group is either \(H'*Z\) or \(H'*\pi_ 1(S)*Z*\cdots*Z\) where \(S\) is a closed surface of non-positive Euler characteristic. The proof of this theorem uses interval exchanges obtained from measured laminations.

57S30 Discontinuous groups of transformations
20F99 Special aspects of infinite or finite groups
57M07 Topological methods in group theory
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
57M05 Fundamental group, presentations, free differential calculus
Full Text: DOI
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