Enumerating finite-index subgroups of two-ended groups

Let $$F$$ be a finite group and $$\psi \colon F \to F$$ a group automorphism.

Question 1: I'm interested in enumerating/characterizing the finite-index subgroups of $$F \rtimes_\psi \mathbb{Z}$$ in a 'closed' way, similar to the description that Goursat's lemma provides for the subgroups of $$F \times \mathbb{Z}$$ (which relies solely on the subgroup structure of $$F$$). I know that the general problem of enumerating or characterizing subgroups of any semidirect product $$F_1 \rtimes F_2$$ is somewhat hopeless, but I hope there exists a 'reasonable' description in $$F \rtimes_\psi \mathbb{Z}$$ in terms of $$F$$ and $$\psi$$.

There's this paper that describes the subgroups of any semidirect product $$G \rtimes_\phi H$$, but I couldn't specialize the results to the case I'm interested in. This question asks for the general case.

Question 2: same as Question 1 but for finite-index subgroups of $$G_1 \ast_F G_2$$, where $$G_1, G_2$$ are finite groups such that $$[G_1:F] = [G_2:F] = 2$$. This paper describes the subgroups of free product with amalgamation, but (again) I couldn't extract any 'down to earth' results. I'm aware that when $$T = \{1\}$$ (that is, $$G$$ is the infinite dihedral group) the enumeration is straightforward.

The motivation for this question is to compute the exponential zeta function for the subgroup growth sequence, that is, the series $$\begin{equation*}g(z) = \exp\left( \sum_{n \geq 0} \frac{a_n(G)}{n}z^n \right)\end{equation*}$$ where $$a_n(G)$$ is the number of subgroups of index $$n$$ in $$G$$ (here $$G$$ is of the form $$F \times \mathbb{Z}$$ or $$G_1 \ast_T G_2$$ as before). Any observations or results for a subclass of these groups are welcome.

• You can use the fact that $\phi$ is of finite order and reduce the problem to the direct product $F\times\mathbb Z$. Oct 12 '21 at 5:39
• Do you have a precise motivation (such as computing the zeta function of the subgroup growth)?
– YCor
Oct 12 '21 at 6:08
• Take $H \leq G \Doteq F \rtimes_{\psi} \mathbb{Z}$ of finite index in $G$. Then $H = (H \cap F) \rtimes_{\psi^k} \mathbb{Z}$ for some $k \ge 0$ depending on $H$. Knowing which subgroups of $F$ are stable under $\psi^k$ for each $k \ge 0$ yields therefore a description of the finite index subgroups of $G$. Oct 12 '21 at 7:36
• @YCor: yes, in fact that's the objective. I'm trying to calculate the exponential zeta function for the subgroup growth sequence. I'm also interested in being able to parametrize these subgroups in some way (that is, not only counting the number of finite-index subgroups but being able to write them down, 'sitting' inside the big group). I feel that this calculation may have been done somewhere... Oct 12 '21 at 21:47

Let $$H \leq G_1 \ast_F G_2$$ be a finite-index subgroup. The action of $$H$$ on the Bass-Serre tree $$T$$ (which is a line) of $$G_1 \ast_F G_2$$ must be cocompact, and the quotient graph provides a decomposition of $$H$$ as a graph of groups $$\mathcal{G}$$. Essentially, two cases can happen:
• The underlying graph of $$\mathcal{G}$$ is a segment. In other words, $$H$$ acts on $$T$$ with a segment $$S$$ as a strict fundamental domain. Vertex-stabilisers (in $$H$$) of interior vertices of $$S$$ have to fix $$S$$ entirely. So $$H$$ is generated by the stabilisers of the endpoints of $$S$$. In other words, $$H$$ is the amalgamated product $$A_1 \ast_B A_2$$ generated by two non-trivial subgroups $$A_1,A_2$$ in two distinct conjugates of $$G_1$$ or $$G_2$$ (not included in the corresponding copy of $$F$$).
• The underlying graph of $$\mathcal{G}$$ is a circle. Then the segment $$S$$ is not a strict fundamental domain: its endpoints lie in the same orbit. In this case, $$H$$ is the HNN extension generated by a subgroup $$A$$ in a conjugate of $$G_1$$ or $$G_2$$ (namely, the stabiliser of the first endpoint of $$S$$) with an infinite-order element $$t$$ (namely, an element that sends the first endpoint of $$S$$ to the second).
For $$F \rtimes_\psi \mathbb{Z}$$, one can argue similarly because it is also an HNN extension of $$F$$.