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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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##### References:
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