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**Covering theory for graphs of groups.**
*(English)*
Zbl 0805.57001

When a group acts without inversions on a simplicial tree, the quotient graph can be regarded as a graph of groups, where the group associated to a vertex or edge is the stabilizer of a preimage vertex or edge in the tree. The first major advance in this theory was by Serre, who developed methods for obtaining presentations of groups acting on trees, as well as transparent explanations for various subgroup theorems. In the work under review, the author extends Serre’s theory by giving a rigorous treatment of covering maps of graphs of groups. The author cites as one of his motivations a covering space argument of Kulkarni for a remarkable theorem of Leighton to the effect that two finite graphs with a common covering have a common finite covering. The author’s work furnishes the algebraic foundations needed to carry out Kulkarni’s ideas.

After giving precise definitions, the author develops a concept of morphism between graphs of groups. A morphism induces a homomorphism between the fundamental groups, and lifts to an equivariant map between the actions on the universal covering trees. The definitions are such that the morphism is an “immersion” if and only if the induced homomorphism is injective and the equivariant lift is injective, and the morphism is a “covering” if and only if the equivariant lift is an isomorphism. The principal result of this first part is the conjugacy theorem, which says that when a subgroup \(H\) of the full group of automorphisms of a tree \(X\) acts without inversions, and \(G_ H\) is the group of covering transformations to \(X\to X/H\), then any subgroup \(\Gamma\) of \(G_ H\) that acts freely on \(X\) is conjugate (by an element of \(G_ H\)) into \(H\).

In the second part of the paper, the author develops the notions of hyperbolic length function, minimal invariant subtree, and minimal action, then specializes to the case of discrete actions (all vertex stabilizers finite). For finitely generated groups, \(\Gamma\) acts discretely on a tree if and only if \(\Gamma\) is virtually free. If \(\Gamma\) acts discretely on a tree, and \(N\) is a finitely generated normal subgroup of \(\Gamma\), then either \(N\) or \(\Gamma/N\) is finite. Finally, if \(\Gamma\) acts discretely and \(\Gamma_ 0\) and \(\Gamma_ 1\) is finitely generated subgroups, then their intersection is finitely generated, and if they are commensurable and not finite then each has finite index in the subgroup they generate.

After giving precise definitions, the author develops a concept of morphism between graphs of groups. A morphism induces a homomorphism between the fundamental groups, and lifts to an equivariant map between the actions on the universal covering trees. The definitions are such that the morphism is an “immersion” if and only if the induced homomorphism is injective and the equivariant lift is injective, and the morphism is a “covering” if and only if the equivariant lift is an isomorphism. The principal result of this first part is the conjugacy theorem, which says that when a subgroup \(H\) of the full group of automorphisms of a tree \(X\) acts without inversions, and \(G_ H\) is the group of covering transformations to \(X\to X/H\), then any subgroup \(\Gamma\) of \(G_ H\) that acts freely on \(X\) is conjugate (by an element of \(G_ H\)) into \(H\).

In the second part of the paper, the author develops the notions of hyperbolic length function, minimal invariant subtree, and minimal action, then specializes to the case of discrete actions (all vertex stabilizers finite). For finitely generated groups, \(\Gamma\) acts discretely on a tree if and only if \(\Gamma\) is virtually free. If \(\Gamma\) acts discretely on a tree, and \(N\) is a finitely generated normal subgroup of \(\Gamma\), then either \(N\) or \(\Gamma/N\) is finite. Finally, if \(\Gamma\) acts discretely and \(\Gamma_ 0\) and \(\Gamma_ 1\) is finitely generated subgroups, then their intersection is finitely generated, and if they are commensurable and not finite then each has finite index in the subgroup they generate.

Reviewer: D.McCullough (Norman)

### Keywords:

groups acting on trees; covering maps of graphs of groups; morphism between graphs of groups; conjugacy theorem; group of automorphisms of a tree; hyperbolic length function; minimal invariant subtree; minimal action; discrete actions; virtually free
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\textit{H. Bass}, J. Pure Appl. Algebra 89, No. 1--2, 3--47 (1993; Zbl 0805.57001)

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### References:

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[7] | Lubotzky, A., Trees and discrete subgroups of Lie groups over local fields, Bull. Amer. Math. Soc., 20, 27-31 (1989) · Zbl 0676.22007 |

[8] | Serre, J. P., Trees (1980), Springer: Springer Berlin |

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[10] | Tits, J., Sur le groupe des automorphismes d’un arbre, (Haefliger, A.; Narasimhan, R., Essays in Topology and Related Topics (1970), Springer: Springer Berlin), 188-211, Memoires dédiés à George de Rham · Zbl 0214.51301 |

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