Covering theory for graphs of groups
type: Item
instance of: scholarly article
title: Covering theory for graphs of groups
zbMATH Open document ID: 0805.57001
DOI: 10.1016/0022-4049(93)90085-8
author: Hyman Bass
published in: Journal of Pure and Applied Algebra
publication date: 1993-10-19T00:00:00Z
review text: 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 discretel…
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 20E08
zbMATH DE Number: 433085
zbMATH Keywords: groups acting on trees
zbMATH Keywords: covering maps of graphs of groups
zbMATH Keywords: morphism between graphs of groups
zbMATH Keywords: conjugacy theorem
zbMATH Keywords: group of automorphisms of a tree
zbMATH Keywords: hyperbolic length function
zbMATH Keywords: minimal invariant subtree
zbMATH Keywords: minimal action
zbMATH Keywords: discrete actions
zbMATH Keywords: virtually free
Wikidata QID: Q105597476
reviewed by: Darryl McCullough
MaRDI profile type: Publication
full work available at URL: 0022-4049(93)90085-8
OpenAlex ID: W2040960421
cites work: Length functions of group actions on \(\Lambda\)-trees
cites work: Uniform Tree Lattices
cites work: Finite and infinite cyclic extensions of free groups
cites work: Finite common coverings of graphs
cites work: Trees and discrete subgroups of Lie groups over local fields
cites work: Q3341041
cites work: Topology of finite graphs
cites work: Q5617560
description: scientific article; zbMATH DE number 433085