A famous theorem of Takahashi states the following:

Theorem. Let $F$ be a free group and let $K_1 \leq K_2 \leq \ldots$ be an ascending chain of finitely generated subgroups of $F$. If the rank of the subgroups in the chain is bounded, then the chain is stationary.

One of the applications of this theorem is to provide a bound for the size of the periodic orbits of a free group endomorphism.

We say that a group $G$ is a Takahashi group if every ascending chain $H_{1}\leq H_{2}\leq \cdots$ of subgroups, each of rank $\leq M$ in $G$, is stationary. In joint work with Vítor Araújo (Salvador da Bahia) and Mihalis Sykiotis (Athens), we have proved that the class of Takahashi groups is closed under finite extensions and finite graphs of groups with virtually polycyclic vertex groups and finite edge groups.

We have shown also that the periodic subgroup is finitely generated for every endomorphism of the fundamental group of a finite graph of groups with finitely generated virtually nilpotent vertex groups and finite edge groups. As a consequence, we can bound the periods for each particular endomorphism of such a group.

With respect to intersections of subgroups, we have proved that a fundamental group $G$ of a finite graph of groups with virtually polycyclic vertex groups and finite edge groups is strongly Howson. More precisely, there exists some constant $M > 0$ such that

${\rm rk}(H_1 \cap H_2) \leq M(n_1-1)(n_2-1)+M$

whenever $H_i \leq G$ and ${\rm rk}(H_i) \leq n_i$ for $i = 1,2$.