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Simplicial approximation and low-rank trees. (English) Zbl 0794.20038
Let \(\Lambda\) be a subgroup of \(\mathbb{R}\). An action without inversion of a finitely generated group \(\Gamma\) on a \(\Lambda\)-tree \(T\) defines a (translation) length function \(\ell\) on \(\Gamma\) taking non-negative values in \(\Lambda\). It is an open problem whether the given action can always be “simplicially approximated”, in the sense that there is a sequence \((\ell_ i)_{i\geq 0}\) of length functions defined by actions of \(\Gamma\) on \(\mathbb{Z}\)-trees, and a sequence \((n_ i)_{i\geq 0}\) of positive integers, such that \(\lim_{t\to\infty}\ell_ i(\gamma)/n_ i= \ell(\gamma)\). In the language of the papers of M. Culler and J. Morgan [Proc. Lond. Math. Soc., III. Ser. 55, 571-604 (1987; Zbl 0658.20021)] and J. W. Morgan and P. B. Shalen [Ann. Math., II. Ser. 120, 401-476 (1984; Zbl 0583.57005)] this says that the projectivized length function defined by the given action is the closure of the set of projectivized length functions defined by simplicial actions. A second question arises in the case that the given action is small: can one take the approximating length functions \(\ell_ i\) to be defined by small simplicial actions?
The main result of the reviewed paper gives affirmative answers to these questions when \(\Gamma\) is finitely presented and \(\Lambda\) has \(\mathbb{Q}\)- rank at most 2, assuming, in the rank-2 case, that the action satisfies the ascending chain condition. In particular, it implies that the second question has an affirmative answer if \(\Lambda\) has \(\mathbb{Q}\)-rank at most 2 and the small subgroups of \(\Lambda\) are finitely generated. The authors also observe that the results remain true if \(\Gamma\) is assumed to be finitely generated, rather than finitely presented, but the given action is assumed to be free.

MSC:
20E08 Groups acting on trees
05C05 Trees
57M15 Relations of low-dimensional topology with graph theory
20F65 Geometric group theory
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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