an:01749644
Zbl 1001.20015
Kapovich, Ilya; Myasnikov, Alexei
Stallings foldings and subgroups of free groups
EN
J. Algebra 248, No. 2, 608-668 (2002).
00083252
2002
j
20E05 57M07 20E07 20E15
Stallings foldings; subgroups of free groups; ascending chains of subgroups
In this long paper, the authors study the subgroups of a free group by using the approach of \textit{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.
Stylianos Andreadakis (Athens)
Zbl 0782.20018