Ergodic theory and free actions of groups on \({\mathbb{R}}\)-trees
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 \textit{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 \textit{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.
scientific article; zbMATH DE number 4107743
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