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Very small group actions on \(\mathbb{R}\)-trees and Dehn twist automorphisms. (English) Zbl 0844.20018
An \(\mathbb{R}\)-tree is a metric space in which any two points are connected by a unique arc (called a geodesic), and this arc is isometric to a real line segment of length equal to the distance between the two points. For a finitely generated group \(G\), the paper is concerned with various analogues of Teichm├╝ller space and its boundary, the group of outer automorphisms of \(G\) being viewed as analogue of the mapping class group. This is made precise by means of appropriate subspaces of the projective space \(SLF(G)\) of what are called translation length functions of small \(G\)-actions on \(\mathbb{R}\)-trees; see e.g. R. Lyndon [Math. Scand. 12, 209-234 (1964; Zbl 0119.26402)] or J. P. Serre, Trees [Springer, Berlin 1980; Zbl 0548.20018] for the notion of length function. The paper is part of a research program aimed at (i) obtaining insight into the structure of \(\text{Out} (G)\) through its induced action on a suitable subspace of \(SLF(G)\) and at (ii) analyzing individual automorphisms by finding fixed points and studying the dynamics of the induced actions on the subspace. To this end, the subspace of \(SLF(G)\), \(VSL(G)\), of what are called translation length functions of very small \(G\)-actions on \(\mathbb{R}\)-trees is introduced; the space \(VSL(G)\), in turn, contains the space \(\text{Free}(G)\) of free \(G\)-actions on \(\mathbb{R}\)-trees.
The first result says that \(VSL(G)\) is compact and that, for \(G\) a free group of rank at least 2, the space \(SLF(G)\) is considerably larger than \(VSL(G)\) (in a certain precise sense). Thereafter a partial answer is obtained to the question whether \(VSL(G)\) is the closure of \(\text{Free} (G)\) in \(SLF(G)\): it is shown that (i) for a group \(G\) which acts freely on an \(\mathbb{R}\)-tree and does not contain a copy of a free abelian group of rank 2, a simplicial \(G\)-action on an \(\mathbb{R}\)-tree lies in \(VSL(G)\) if and only if it is in the closure of \(\text{Free} (G)\) and that (ii) if \(G\) is in addition free nonabelian, a simplicial \(G\)-action on an \(\mathbb{R}\)-tree lies in \(VSL(G)\) if and only if it is a limit of free simplicial actions. This result has been extended thereafter by M. Bestvina and M. Feighn [unpublished]; they proved that when \(G\) is free of rank at least 2, \(VSL(G)\) is indeed the closure of \(\text{Free} (G)\) and that, furthermore, this space also coincides with the closure of the space of free simplicial \(G\)-actions on \(\mathbb{R}\)-trees, studied by M. Culler and K. Vogtman [Invent. Math. 84, 91-119 (1986; Zbl 0589.20022)]. For \(G\) a free group of rank at least 2, thereafter a precise geometric description of the dynamics of the homeomorphism of \(SLF(G)\) induced by a proper Dehn twist is given, too complicated to be reproduced here. It is finally shown that this result, in turn, entails a certain uniqueness property of proper Dehn twist representations. By means of this uniqueness result, the authors solved the conjugacy problem for Dehn twist automorphisms of free groups elsewhere.

20E08 Groups acting on trees
20E36 Automorphisms of infinite groups
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
20E05 Free nonabelian groups
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20F65 Geometric group theory
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