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 Teichmüller 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 This result has been extended thereafter by M. Bestvina and M. Feighn [unpublished]; they proved that when $$G$$ is free of rank at least 2, $$VSL(G)$$ is indeed the closure of $$\text{Free} (G)$$ and that, furthermore, this space also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.

MSC:

 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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