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Geometric group actions on trees. (English) Zbl 0878.20019
The concept of a geometric action of a group \(G\) on an \(R\)-tree has been used by many authors, with various meanings. One of its main uses is to replace (or approximate) a given action on an \(R\)-tree by one which is simpler to analyze while keeping control on the edge stabilizers. An action is considered to be geometric when it arises from the following kind of construction. Start with a measured foliation \(\mathcal F\) on a finite complex \(\Sigma\) with \(\pi_1(\Sigma)=G\). It lifts to a measured foliation \(\widetilde{\mathcal F}\) in the universal covering \(\widetilde\Sigma\). Associated to this is the leaf space made Hausdorff. That is, a pseudometric is defined on \(\widetilde\Sigma\) by letting the pseudodistance between two points be the infimum of the transverse measures of the arcs connecting them, and then points of \(\widetilde\Sigma\) at pseudodistance 0 are identified with each other. The resulting metric space \(T(\widetilde{\mathcal F})\) is an \(R\)-tree with a natural isometric action of \(G\). In the authors’ definition, an extra condition is added to ensure that the leaves of \(\widetilde{\mathcal F}\) are closed: every compact arc transverse to \(\widetilde{\mathcal F}\) may be subdivided into finitely many subintervals that are mapped isometrically into \(T(\widetilde{\mathcal F})\). Secondly, to allow the definition to apply to groups \(G\) that are finitely generated but not necessarily finitely presented, the authors replace the universal covering \(\widetilde\Sigma\to\Sigma\) by a normal covering \(\overline\Sigma\to\Sigma\) having group of covering transformations equal to \(G\). In the formal definition, \(\Sigma\) is 2-dimensional and the foliation has one of two simple forms on each 2-simplex: there must be an edge \(e\) such that either each leaf meets \(e\) transversely or each leaf is parallel to \(e\).
To characterize geometric actions of a finitely generated group \(G\), the authors use the notion of strong limit due to Gillet-Shalen. Roughly speaking, a sequence of trees \(T_n\) is said to converge strongly to a tree \(T\) when any finite subtree of \(T\) may be lifted isometrically, and equivariantly with respect to some finite subset of \(G\), to \(T_n\) for all sufficiently large \(n\). The characterization applies to an \(R\)-tree \(T\) with a minimal action of a \(G\) (that is, no proper subtree is invariant under \(G\)): such an action is nongeometric if and only if it is a nontrival strong limit of geometric actions of \(G\) on \(R\)-trees. That is, a geometric action can only be a strong limit of a sequence that is eventually stationary. If \(G\) is also finitely presented, then it has a geometric action on an \(R\)-tree \(T'\) such that the edge stabilizers for \(T'\) are subgroups of edge stabilizers for \(T\). Another result is that if \(T\) has a geometric action with no global fixed point, and \(T_{\text{min}}\) is the smallest nonempty invariant subtree, then \(T_{\text{min}}\) is closed in \(T\) and the restriction of the action to \(T_{\text{min}}\) is geometric. In the case of simplicial actions, a minimal simplicial action is geometric if and only if all edge groups are finitely generated. Additional results concern the orbits of branch points, abelian actions, and actions with trivial edge stabilizers.

20E08 Groups acting on trees
57M07 Topological methods in group theory
57M60 Group actions on manifolds and cell complexes in low dimensions
20F65 Geometric group theory
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