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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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