zbMATH — the first resource for mathematics

Quasiconvexity and a theorem of Howson’s. (English) Zbl 0869.20023
Ghys, É. (ed.) et al., Group theory from a geometrical viewpoint. Proceedings of a workshop, held at the International Centre for Theoretical Physics in Trieste, Italy, 26 March to 6 April 1990. Singapore: World Scientific. 168-176 (1991).
The concept, due to Gromov, of a “quasiconvex” subgroup of a finitely generated group is defined via the usual word metric on the Cayley graph of the group. Such subgroups are shown here to be finitely generated. Geometrical ideas are applied to yield a “leisurely” proof of Howson’s theorem, namely, that the intersection of two finitely generated subgroups of a free group is finitely generated. The paper is pleasant to read.
Note that it is not much more than an exercise to show that a finite extension of a group with the “finitely generated intersection property”, again has that property: – à propos of a remark in the paper.
For the entire collection see [Zbl 0809.00017].

20F65 Geometric group theory
20E07 Subgroup theorems; subgroup growth
20E05 Free nonabelian groups