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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