Subgroups of Cyclically Amalgamated Free Products

TL;DR

This paper investigates the subgroup structure of cyclically amalgamated free products, establishing conditions under which subgroups are free products of cyclic groups or related quotients, and revisits a conjecture about groups with all infinite index subgroups free.

Contribution

It provides new results on the subgroup structure of cyclically amalgamated free products, especially regarding n-free products of cyclics and subgroup classifications.

Findings

If H1 and H2 are 3-free products of cyclics of rank ≥ 3, then G is also a 3-free product.

If H1 and H2 are 4-free products of cyclics of rank ≥ 4, then every 4-generated subgroup of G has a specific free product structure.

Revisits and elaborates on proofs related to a conjecture about groups with all infinite index subgroups being free.

Abstract

Given a group $G = H_1 \ast_A H_2$ which is the free product of two finitely generated groups $H_1$ and $H_2$ with amalgamation over a cyclic subgroup $A$ which is malnormal in $G$, we study relations between the structure of its subgroups and the structure of the group $G$ itself. Firstly, we show that if $H_1$ and $H_2$ are 3-free products of cyclics of rank $\ge 3$ then $G$ is also a 3-free product of cyclics. Secondly, we prove that if $H_1$ and $H_2$ are 4-free products of cyclics of rank $\ge 4$ then every 4-generated subgroup of $G$ is a free product of $\le 4$ cyclics or a 1-relator quotient of a free product of four cyclic groups. Here a group is called an $n$-free product of cyclics if every $n$-generated subgroup is a free product of $\le n$ cyclic groups. These results are based on ubiquitous applications of the Nielsen method for amalgamated free products which we recall carefully. Lastly, given an infinite, finitely presented group which is not free, but all of its infinite index subgroups are free, a well-known conjecture says that it is isomorphic to a surface group. We revisit and elaborate on predominantly group theoretic proofs of this conjecture for cyclically amalgamated products as above, as well as for certain HNN extensions.