Graphs of actions on \(\mathbf R\)-trees. (English) Zbl 0802.05044

Levitt, Gilbert

Comment. Math. Helv. 69, No. 1, 28-38 (1994).

The main results are the following theorems:
Let \(G\) be a finitely generated group acting minimally on an R-tree \(T\). If the action is free, then the associated metric space \(\widehat{T/G}\) is a finite graph.
Let \(G\) be a finitely generated group acting on an R-tree \(T\), with length function \(l\).
(1) There exists \(\varepsilon_ 0>0\) such that \(G(\varepsilon)\) is independent of \(\varepsilon\) for \(\varepsilon<\varepsilon_ 0\). Denote it by \(G_ 0\).
(2) The quotient \(G/G_ 0\) is free of rank \(m(l)\).
(3) The space \(T/G\) has the homotopy type of a wedge of \(m(l)\) circles. It has a universal covering: the R-tree \(\widehat{T/G_ 0}\).
Let \(G\) be a finitely generated group acting freely and minimally on an R-tree \(T\). The graph \(\widehat{T/G}\) is homeomorphic to a segment iff, given \(g\in G\) and \(\varepsilon>0\), there exist \(g_ 1\) and \(g_ 2\) such that \(g= g_ 1 g_ 2\) and \(\max(l(g_ 1),l(g_ 2))<{1\over 2} l(g)+\varepsilon\).

Reviewer: U.Baumann (Dresden)

MSC:

05C25	Graphs and abstract algebra (groups, rings, fields, etc.)
05C05	Trees

Keywords:

R-tree; metric space; homotopy; universal covering