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Ergodic theory and free actions of groups on \({\mathbb{R}}\)-trees. (English) Zbl 0676.57001
The author answered the question: which amalgamated free products \(G=G_ 0*_{{\mathbb{Z}}} G_ 1\) with amalgamating subgroup an infinite cyclic group \({\mathbb{Z}}\) can act freely on an \({\mathbb{R}}\)-tree? Put \({\mathbb{Z}}=<\gamma >\), \(j_ i(\gamma)=\gamma_ i\), \(j_ i: {\mathbb{Z}}\hookrightarrow G_ i\), \(i=0,1\), and call a reduced word in a free group with basis \(e_ 1,...,e_ k\) “quadratic”, if \(e_ i^{\pm 1}\) occurs at most twice in it.
Theorem 1: If \(G_ 0*_{{\mathbb{Z}}} G_ 1\) acts freely on an \({\mathbb{R}}\)- tree, either \(\gamma_ i\) generates a free factor of \(G_ i\) for \(i=0\) or \(i=1\) (trivial case), or \(G_ i=H_ i*F_ i\), \(F_ i\) free, and \(\gamma_ i\in F_ i\) quadratic for \(i=0,1\). - If the \(G_ i\) are finitely presented, one has \(G\cong H_ 0*G'*H_ 1\), \(H_ i\) a free factor of \(G_ i\), and \(G'\) a free product of a surface group and a free group.
Such groups do act freely on an \({\mathbb{R}}\)-tree, if one excludes the 3 smallest non-orientable surface groups [see the author and P. Shalen, Free actions of surface groups on \({\mathbb{R}}\)-trees (to appear)]. The problem is solved by producing a fixed point for an assumed free action of G in all cases save the “small” ones mentioned in Theorem 1.
The theory of codimension-1 measured laminations developed by the author and P. Shalen [Ann. Math., II. Ser. 127, No. 2, 403-456 (1988; Zbl 0656.57003); ibid., No. 3, 457-519 (1988; Zbl 0661.57004)] is used together with a sophisticated argument implying the first ergodic theorem.
Reviewer: G.Burde

MSC:
57M15 Relations of low-dimensional topology with graph theory
57M05 Fundamental group, presentations, free differential calculus
57N10 Topology of general \(3\)-manifolds (MSC2010)
57R30 Foliations in differential topology; geometric theory
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References:
[1] Halmos, P.:Lectures on ergodic theory. Publ. Math. Soc. Japan3, Tokyo, 1956 · Zbl 0073.09302
[2] Lyndon, R., Shupp, P.:Combinatorial group theory. (Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 89. Berlin-Heidelberg-New York: Springer 1977
[3] Morgan, J., Shalen, P.: Degenerations of hyperbolic structures, II: Measured laminations in 3-manifolds. Ann. Math.127, 403-456 (1988) · Zbl 0656.57003
[4] Morgan, J., Shalen, P.: Degenerations of hyperbolic structures, III: Actions of 3-manifold groups on trees and Thurston’s compactness theorem. Ann. Math.127, 457-519 (1988) · Zbl 0661.57004
[5] Morgan, J., Shalen, P.: Free actions of surface groups on ?-trees (to appear)) · Zbl 0726.57001
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