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Large scale geometry of commutator subgroups. (English) Zbl 1195.20046
Summary: Let \(G\) be a finitely presented group, and \(G'\) its commutator subgroup. Let \(C\) be the Cayley graph of \(G'\) with all commutators in \(G\) as generators. Then \(C\) is large scale simply connected. Furthermore, if \(G\) is a torsion-free nonelementary word-hyperbolic group, \(C\) is one-ended. Hence (in this case), the asymptotic dimension of \(C\) is at least 2.
MSC:
20F65 Geometric group theory
20F67 Hyperbolic groups and nonpositively curved groups
57M07 Topological methods in group theory
20F05 Generators, relations, and presentations of groups
20F12 Commutator calculus
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
55M10 Dimension theory in algebraic topology
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