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*-groups, graphs, and bases. (English) Zbl 0743.20019
Topology and combinatorial group theory, Proc. Fall Foliage Topology Semin., New Hampshire/UK 1986-88, Lect. Notes Math. 1440, 186-191 (1990).
[For the entire collection see Zbl 0701.00019.]
In 1949 Marshall Hall proved the now well-known theorem: If $$F$$ is a finitely generated free group, $$H_ 0$$ a finitely generated subgroup, and $$x_ 1,\dots,x_ m$$ are any elements of $$F$$ which are not in $$H_ 0$$, then there exists a subgroup $$H^*$$ of $$F$$ satisfying i) $$H^*$$ has finite index in $$F$$, ii) $$H_ 0$$ is a free factor of $$H^*$$, and iii) $$x_ 1,\dots,x_ m\not\in H^*$$. A subgroup $$H^*$$ which satisfies i) and ii) is called a *-group for $$H_ 0$$. The paper provides an effective method for constructing all *-groups for a given subgroup $$H_ 0$$. This method uses the folding and immersion techniques for graphs developed by Stallings.

##### MSC:
 20E05 Free nonabelian groups 20E07 Subgroup theorems; subgroup growth 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 57M07 Topological methods in group theory 20F05 Generators, relations, and presentations of groups