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Automorphisms of graph groups. (English) Zbl 0682.20022
Given a graph \(\Gamma=(V,E)\), the graph group \(F\langle\Gamma\rangle\) is the group with presentation \(\langle V\mid [E]\rangle\), where \([E]\) denotes the set of commutators \(\{[a,b]\mid\{a,b\}\in E\}\). The graph group \(F\langle\Gamma\rangle\) is modeled to be a group analog of the graph algebra K(\(\Gamma)\) generated as a free associative algebra on \(V\) modulo the ideal generated by commutators \([a,b]=ab-ba\) for \(\{a,b\}\in E\). Graph algebras were first studied by Kim, Makar-Limanov, Neggers and Roush. In this paper the author studies analogs in \(F\langle\Gamma\rangle\) of the Nielsen automorphisms of free groups, which reduce to Nielsen automorphisms in the case \(\Gamma=(V,\emptyset)\). The Centralizer Theorem describes the centralizer of elements \(u\) of \(F\langle\Gamma\rangle\). It follows from this description that the centralizer of a graph group is itself a graph group, which is a very nice result even if expected. A strong conjecture of this paper is that the elementary automorphisms (i.e., the analogs of the Nielsen automorphisms) generate the group of automorphisms of \(F\langle\Gamma\rangle\). The special cases proven are \(\Gamma=T\) a tree and \(\Gamma\) is star two-connected containing no triangles or squares. In arriving at these proofs the author develops a generalization of a theorem due to Humphreys on generating sets of free groups. Several additional useful lemmas and propositions provide further computational insights and demonstrate that a theory of graph groups, just as a theory of graph algebras, has a bright present as well as a bright future.

MSC:
20F05 Generators, relations, and presentations of groups
20E36 Automorphisms of infinite groups
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20F28 Automorphism groups of groups
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