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Free actions of surface groups on \({\mathbb{R}}\)-trees. (English) Zbl 0726.57001

The authors make progress on the question of which groups act freely (by isometries) on \({\mathbb{R}}\)-trees. The collection of such groups is closed under free products, and contains all subgroups of \({\mathbb{R}}\) itself (in particular, the finitely generated free abelian groups). R. C. Alperin and K. N. Moss [J. Lond. Math. Soc., II. Ser. 31, 55-68 (1985; Zbl 0571.20031)] gave the first examples which are not free products of subgroups of \({\mathbb{R}}\). Their examples are not finitely generated. In the paper under review, the authors provide finitely generated examples, by proving that when G is the fundamental group of a closed surface, it acts freely as isometries on some \({\mathbb{R}}\)-tree if and only if the surface is not one of the three surfaces which are nonorientable and have Euler characteristic \(\geq -1\). W. Parry also proved this result independently for the fundamental groups of nonorientable surfaces.
The method of proof (in the case of a hyperbolic surface group, the other cases being easy) is to show that the surface contains a measured lamination all of whose complementary components are simply-connected. Then, the action of the surface group on the dual tree to the preimage of the lamination in the universal cover is the desired free action. For the exceptional cases, showing that no action exists is easy for the projective plane and Klein bottle, but a clever argument is needed to show that any action of the fundamental group of the connected sum of three projective planes has a fixed point.
The authors ask whether any finitely generated group which acts freely on an \({\mathbb{R}}\)-tree is a free product of non-exceptional surface groups and finitely generated free abelian groups.

MSC:

57M07 Topological methods in group theory
57S99 Topological transformation groups
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)

Citations:

Zbl 0571.20031
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