Weak Z-structures for some combinatorial group constructions

Bestvina [1] introduced the notion of a (weak) $Z$-structure and (weak) $Z$-boundary for a torsion-free group, motivated by the notion of boundary for hyperbolic and $CAT\left(0\right)$ groups. Since then, some classes of groups have been shown to admit a (weak) $Z$-structure (see [5], [20], [22] for example); in fact, in all cases these groups are semistable at infinity and happen to have a pro-(finitely generated free) fundamental pro-group. The question whether or not every type $F$ group admits such a structure remains open. In [33] it was shown that the property of admitting such a structure is closed under direct products and free products. Our main results are as follows.

THEOREM: Let G be a torsion-free and semistable at infinity finitely presented group with a pro-(finitely generated free) fundamental pro-group at each end. If G has a finite graph of groups decomposition in which all the groups involved are of type $F$ and inward tame (in particular, if they all admit a weak $Z$-structure) then G admits a weak $Z$-structure.

COROLLARY: The class of those 1-ended and semistable at infinity torsion-free finitely presented groups which admit a weak $Z$-structure and have a pro-(finitely generated free) fundamental pro-group is closed under amalgamated products (resp. HNN-extensions) over finitely generated free groups.

On the other hand, given a finitely presented group G and a monomorphism $\phi :G⟶G$, we may consider the ascending HNN-extension $G{⁎}_{\phi }=〈G,t;t^{-1}gt=\phi \left(g\right),g\in G〉$. The results in [26] together with the Theorem above yield the following:

PROPOSITION: If a finitely presented torsion-free group G is of type $F$ and inward tame, then any (1-ended) ascending HNN-extension $G{⁎}_{\phi }$ admits a weak $Z$-structure.

In the particular case $\phi \in Aut\left(G\right)$, this ascending HNN-extension corresponds to a semidirect product $G{⋊}_{\phi }Z$, and it has been shown in [18] that if G admits a $Z$-structure then so does $G{⋊}_{\phi }Z$.