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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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