## Stallings foldings and subgroups of free groups.(English)Zbl 1001.20015

In this long paper, the authors study the subgroups of a free group by using the approach of J. R. Stallings, who introduced the notion of foldings of graphs [in Arboreal group theory, Publ., Math. Sci. Res. Inst. 19, 355-368 (1991; Zbl 0782.20018)]. In their own words “they re-cast in a more combinatorial and computational form the topological approach of J. Stallings to the study of subgroups of free groups”. For this, they give a detailed, selfcontained, elementary and comprehensive treatment of the used approach. They also include “complete and independent proofs of most basic facts, a substantial number of explicit examples and a wide assortment of possible applications”. In doing so they reprove many classical well-known, or folklore results about the subgroup structure of free groups. For example they prove the Takahasi-Higman theorem on ascending chains of subgroups of a free group. More precisely: Let $$F=F(X)$$ be a free group of finite rank. Let $$M\geq 1$$ be an integer. Then every strictly ascending chain of subgroups of $$F(X)$$ of rank at most $$M$$ terminates. Concluding we can say that this paper is a good account of the subgroup structure of free groups in this setting. The reference list contains 48 items.

### MathOverflow Questions:

Membership to double cosets in free groups

### MSC:

 20E05 Free nonabelian groups 57M07 Topological methods in group theory 20E07 Subgroup theorems; subgroup growth 20E15 Chains and lattices of subgroups, subnormal subgroups

Zbl 0782.20018
Full Text:

### References:

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