Algorithmic recognition of infinite cyclic extensions

Algorithmic behavior of groups has been a very fundamental concern in Combinatorial and Geometric Group Theory since the very beginning of this branch of Mathematics in the early 1900’s. The famous three problems stated by Max Dehn in 1911 are prototypical examples of this fact: the Word, Conjugacy, and Isomorphism problems have been very influential in the literature along these last hundred years. Today, these problems (together with a great and growing collection of variations) are the center of what is known as Algorithmic Group Theory.

Dehn’s Isomorphism Problem is probably the paradigmatic example of what is popularly understood as “algorithmic recognition of groups”. Namely, let $\mathcal{G}_{\mathsf{fp}}$ be the family of finite presentations of groups. Then (with the usual abuse of notation of denoting in the same way a presentation and the presented group):

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  Isomorphism problem [ $\operatorname{IP}$ ]: given two finite presentations $G_{1},G_{2}\in\mathcal{G}_{\mathsf{fp}}$, decide whether they present isomorphic groups, $G_{1}\simeq G_{2}$, or not.

It is well known that, in this full generality, Dehn’s Isomorphism Problem is unsolvable, see for example [27]. So, a natural next step is to study what happens when we restrict the inputs to a certain subfamily $\mathcal{H}\subseteq\mathcal{G}_{\mathsf{fp}}$:

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  Isomorphism problem within $\mathcal{H}$ [ $\operatorname{IP}(\mathcal{H})\,]$: given two finite presentations $H_{1},H_{2}\in\mathcal{H}$, decide whether they present isomorphic groups, $H_{1}\simeq H_{2}$, or not.

Since we are interested in groups, we will only consider families of presentations closed by isomorphism; in this way, the problems considered are actually about groups (although represented by finite presentations). The literature is full of results solving the isomorphism problem for more and more such subfamilies $\mathcal{H}$ of $\mathcal{G}_{\mathsf{fp}}$, or showing its unsolvability even when restricted to smaller and smaller subfamilies $\mathcal{H}$.

Another recognition aspect is that of deciding whether a given group satisfies certain property, i.e., whether it belongs to a certain previously defined family. For two arbitrary subfamilies $\mathcal{H},\mathcal{G}\subseteq\mathcal{G}_{\mathsf{fp}}$, we define the:

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  (Family) Membership problem for $\mathcal{H}$ within $\mathcal{G}$ [ $\operatorname{MP}_{\mathcal{G}}(\mathcal{H})$ ]: given a finite presentation ${G\in\mathcal{G}}$, decide whether $G\in\mathcal{H}$ or not.

If $\mathcal{H}\subseteq\mathcal{G}$ and $\operatorname{MP}_{\mathcal{G}}(\mathcal{H})$ is decidable we will also say that the inclusion $\mathcal{H}\subseteq\mathcal{G}$ is decidable. When the considered ambient family is the whole family of finitely presented groups (i.e., $\mathcal{G}=\mathcal{G}_{\mathsf{fp}}$) we will usually omit any reference to it and simply talk about the membership problem for $\mathcal{H}$, denoted $\operatorname{MP}(\mathcal{H})$.

A classic undecidability result due to Adian [1, 2] and Rabin [29] (see also [27]) falls into this scheme. Namely, when $\mathcal{H}$ is a Markov subfamily (i.e., a nonempty subfamily $\emptyset\neq\mathcal{H}\subseteq\mathcal{G}_{\mathsf{fp}}$ such that the subgroups of groups in $\mathcal{H}$ do not completely cover $\mathcal{G}_{\mathsf{fp}}$), then $\operatorname{MP}(\mathcal{H})$ is not decidable. This turns out to include the impossibility of deciding membership for countless well known families of finitely presented groups (e.g., trivial, finite, abelian, nilpotent, solvable, free, torsion-free, simple, automatic, etc).

Note that $\operatorname{MP}(\mathcal{H})$ being decidable is the same as saying that $\mathcal{H}$ is a recursive set of finite presentations. And, even when $\operatorname{MP}(\mathcal{H})$ is not decidable, we can still ask for a recursive enumeration of the elements in $\mathcal{H}$:

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  (Family) Enumeration problem for $\mathcal{H}$ [ $\operatorname{EP}(\mathcal{H})$ ]: enumerate all the elements in $\mathcal{H}$.

In many cases the considered subfamily $\mathcal{H}\subseteq\mathcal{G}_{\mathsf{fp}}$ entails a concept of “good” or “standard” presentation $\mathcal{S}\subseteq\mathcal{H}$, for the groups presented. For example, if $\mathcal{H}$ is the family of (finite presentations for) braid groups $\{B_{n}\mid n\geqslant 2\}$, we can define the set of standard presentations $\mathcal{S}\subseteq\mathcal{H}$ to be those of the form

in this case, the family enumeration problem for $\mathcal{S}$ consists, on input an arbitrary finite presentation presenting a braid group, to compute its (unique) standard one for it, i.e., to recognize the number of strands $n$.

Let $G$ and $Q$ be arbitrary groups. We say that $G$ is a group extension by $Q$ (or a $Q$-extension) if $G$ can be homomorphically mapped onto $Q$, i.e., if there exists a normal subgroup $H\trianglelefteqslant G$ such that the quotient $G/H$ is isomorphic to $Q$. Of course, this situation gives rise to the short exact sequence

for some group $H$, and we will say that $G$ is $*\operatorname{-by-}Q$. One also says that such $H$ is a base group for the extension, and that $G$ is an extension of $H$ by $Q$; so, if we want to specify the base group we will say that $G$ is $H\operatorname{-by-}Q$.

We remark that a given group extension by $Q$ may admit many, even non isomorphic, different base groups (see 8.2).

If $H$ is the only (as subset) normal subgroup of $G$ with quotient $G/H$ isomorphic to $Q$, then we say that the $Q$-extension is unique, or that $G$ is a unique extension by $Q$; in the same vein as before, we will say that $G$ is $!\operatorname{-by-}Q$ (or that $G$ is $!H\operatorname{-by-}Q$, if we want to specify who is the unique normal subgroup).

It will be convenient to extend this notation allowing to replace the groups $H$ and $Q$ by any group property (which we will usually write in $\mathsf{sans}$ typeface). Concretely, given two properties of groups, $\mathsf{P}_{1}$, $\mathsf{P}_{2}$, we say that a group $G$ is $\mathsf{P}_{1}\operatorname{-by-}\mathsf{P}_{2}$ (resp., $!\mathsf{P}_{1}\operatorname{-by-}\mathsf{P}_{2}$) if it is $H\operatorname{-by-}Q$ (resp., $!H\operatorname{-by-}Q$) for certain groups $H$ satisfying $\mathsf{P}_{1}$, and $Q$ satisfying $\mathsf{P}_{2}$. In this way we can easily refer to families of group extensions in terms of the behavior of their base and quotient groups. So, for example, a group $G$ is $\mathsf{fg}\operatorname{-by-}\mathbb{Z}^{r}$ if it is $H\operatorname{-by-}\mathbb{Z}^{r}$ for some finitely generated group $H$. And it is $!\mathsf{fg}\operatorname{-by-}\mathbb{Z}^{r}$ if it is $!H\operatorname{-by-}\mathbb{Z}^{r}$ for some finitely generated group $H$; i.e., if it has a unique normal subgroup with quotient isomorphic to $\mathbb{Z}^{r}$, which happens to be finitely generated (not to be confused with $G$ having a unique finitely generated normal subgroup whose quotient is isomorphic to $\mathbb{Z}^{r}$—and possibly some others infinitely generated as well).

When we need to add extra assumptions (i.e., satisfying some property $\mathsf{P}$) on the elements of certain family $\mathcal{G}$, we will denote the new family $[\mathcal{G}]_{\mathsf{P}}$. For example, $[\mathsf{abelian}]_{\mathsf{fg}}$ denotes the family of finitely generated abelian groups, while $[*\operatorname{-by-}\mathbb{Z}]_{\mathsf{fp}}$ denotes the family of finitely presented extensions by $\mathbb{Z}$.

Recognizing $\mathbb{Z}^{r}$-extensions (more concretely, solving the membership problem for the families $*\operatorname{-by-}\mathbb{Z}^{r}$ and $!\operatorname{-by-}\mathbb{Z}^{r}$) turns out to be very easy. Recall that, for a finitely generated group $G$, its abelianization is always of the form $G^{\operatorname{ab}}=\mathbb{Z}^{r}\oplus T$, where $r$ is a nonnegative integer and $T$ is a finite abelian group (both canonically determined by $G$ and called, respectively, the (first) Betti number of $G$, denoted $r=\operatorname{b}(G)$, and the abelian torsion of $G$).

Clearly, for a finitely generated group $G$, $\operatorname{b}(G)$ is the maximum rank for a free-abelian quotient of $G$. Hence, we have the following straightforward characterizations.

Note that one can easily compute the Betti number of any group given by a finite presentation: just abelianize it (i.e., add as relators the commutators of any pair of generators in the presentation) and then apply the Classification Theorem for finitely generated abelian groups, which is clearly algorithmic. Thus, Lemma 2.1 immediately implies the decidability of the membership problem for these families of groups.

Let us denote by $\pi^{\operatorname{ab}}\colon G\twoheadrightarrow G^{\operatorname{ab}}$ the abelianization map, by $\pi^{\operatorname{ab}}_{*}\colon G\twoheadrightarrow\mathbb{Z}^{\operatorname{b}(G)}$ the abelianization map followed by the canonical projection onto the free-abelian part $\mathbb{Z}^{\operatorname{b}(G)}$, and let us also write $G^{\operatorname{ab}}_{*}=\pi^{\operatorname{ab}}_{*}(G)$. We collect here some elementary properties of the first Betti number which will be useful later.

In addition, note that the kernels of both $\pi^{\operatorname{ab}}$ and $\pi^{\operatorname{ab}}_{*}$ are fully characteristic subgroups of $G$ (i.e., invariant under endomorphisms of $G$). Hence, every $\phi\in\operatorname{End}(G)$ (resp., every $\phi\in\operatorname{Aut}(G)$) canonically determines endomorphisms $\phi^{\operatorname{ab}}\in\operatorname{End}(G^{\operatorname{ab}})$ and ${\phi^{\operatorname{ab}}_{*}\in\operatorname{End}(G^{\operatorname{ab}}_{*})}$ (resp., automorphisms $\phi^{\operatorname{ab}}\in\operatorname{Aut}(G^{\operatorname{ab}})$ and $\phi^{\operatorname{ab}}_{*}\in\operatorname{Aut}(G^{\operatorname{ab}}_{*})$). Of course, after choosing an abelian basis for $\mathbb{Z}^{r}$, where $r=\operatorname{b}(G)$, $\phi^{\operatorname{ab}}_{*}$ can be thought of as an $r\times r$ square matrix over the integers (with determinant $\pm 1$ if $\phi$ is an automorphism of $G$). In the following section we will relate certain properties of a $\mathbb{Z}$-extension with properties of its defining automorphism $\phi$.

We will concentrate now on infinite cyclic extensions, concretely in the family $*\operatorname{-by-}\mathbb{Z}$ and its subfamily $!\operatorname{-by-}\mathbb{Z}$. Let us describe them in a different way: since $\mathbb{Z}$ is a free group, every short exact sequence of the form $1\to H\to G\to\mathbb{Z}\to 1$ splits, and so $G$ is a semidirect product of $H$ by $\mathbb{Z}$; namely $G\simeq H\rtimes_{\alpha}\mathbb{Z}$, for some $\alpha\in\operatorname{Aut}(H)$. Let us recall this well known construction in order to fix our notation.

Given an arbitrary group $H$ and an automorphism $\alpha\in\operatorname{Aut}(H)$, define the semidirect product of $H$ by $\mathbb{Z}$ determined by $\alpha$ as the group $H\rtimes_{\alpha}\mathbb{Z}$ with underlying set $H\times\mathbb{Z}$ and operation given by

for all $h,k\in H$, and $m,n\in\mathbb{Z}$. Of course, $h\mapsto(h,0)$ is a natural embedding of $H$ in $H\rtimes_{\alpha}\mathbb{Z}$, and we then have the natural short exact sequence

Therefore, $H\rtimes_{\alpha}\mathbb{Z}$ belongs to the family $*\operatorname{-by-}\mathbb{Z}$. Recall that we can have $H\rtimes_{\alpha}\mathbb{Z}\simeq K\rtimes_{\beta}\mathbb{Z}$, with $H=K$ but $\alpha\neq\beta$; or even with $H\not\simeq K$. We discuss this phenomena in Section 8 (see 8.3 and 8.2, respectively).

In the opposite direction, assume that $G$ is in the family $*\operatorname{-by-}\mathbb{Z}$. Choose a homomorphism onto $\mathbb{Z}$, say $\rho\colon G\twoheadrightarrow\mathbb{Z}$, and consider the short exact sequence given by

where $H=\ker\rho\trianglelefteqslant G$. Choose and denote by $t$ a preimage in $G$ of any of the two generators of $\mathbb{Z}$ (note that choosing such $t$ is equivalent to choosing a split homomorphism for $\rho$). Now consider the conjugation by $t$ in $G$, say $\gamma_{t}\colon G\to G$, $g\mapsto tgt^{-1}$, and denote by $\alpha\in\operatorname{Aut}(H)$ its restriction to $H$ (note that $\gamma_{t}$ is an inner automorphism of $G$, but $\alpha$ may very well not be inner as an automorphism of $H$). By construction, we have

for every $h\in H$. At this point, it is clear that every element from $G$ can be written in a unique way as $ht^{k}$, for some $h\in H$ and $k\in\mathbb{Z}$. And—from (3)—the operation in $G$ can be easily understood by thinking that $t$ (respectively, $t^{-1}$) jumps to the right of elements from $H$ at the price of applying $\alpha$ (respectively, $\alpha^{-1}$):

This is, precisely, the multiplicative version of (1). Hence, $G\simeq H\rtimes_{\alpha}\mathbb{Z}$, the semidirect product of $H$ by $\mathbb{Z}$ determined by $\alpha$.

From this discussion it follows easily that, for any presentation of $H$, say $H=\langle\,X\!\mid\!R\,\rangle$, and any $\alpha\in\operatorname{Aut}(H)$, the semidirect product $G=H\rtimes_{\alpha}\mathbb{Z}$ admits a presentation of the form

Note that (5) is a finite presentation if and only if the initial presentation for $H$ was finite. So, a group $G$ admits a finite presentation of type (5) if and only if $G$ is $\mathsf{fp}\operatorname{-by-}\mathbb{Z}$. This provides the notion of standard presentation in this context.

The previous discussion provides the following alternative descriptions for the family of finitely presented $\mathbb{Z}$-extensions. For any group $H$, we have

and then,

The strictness of the second inclusion in (6) is a direct consequence of 5.2, while the strictness of the first one is proved below (we thank Conchita Martínez for pointing out the candidate group (7) in the subsequent proof).

Recall that the family of unique $\mathbb{Z}$-extensions (i.e., groups having a unique normal subgroup with quotient $\mathbb{Z}$) which are finitely presented is denoted $[!\operatorname{-by-}\mathbb{Z}]_{\mathsf{fp}}$.

As seen in 2.1, the family $[!\operatorname{-by-}\mathbb{Z}]_{\mathsf{fp}}$ consists precisely of those groups $G$ in $[*\operatorname{-by-}\mathbb{Z}]_{\mathsf{fp}}$ such that $\operatorname{b}(G)=1$. For finitely generated base groups $H$, LABEL:prop:derangedcharacterization gives a quite simple characterization of this unicity condition in terms of the defining automorphism $\alpha$. We first need a convenient description of the abelianization of a $\mathbb{Z}$-extension.

This last result, combined with 2.1, provides a computable characterization for automorphisms defining $!\mathsf{fg}\operatorname{-by-}\mathbb{Z}$ groups.

In particular, every automorphism of a group $H$ with $\operatorname{b}(H)=0$, i.e., with $|H^{\operatorname{ab}}|<\infty$, is (trivially) deranged. Note also that derangedness is, in fact, a property of outer automorphisms. The sets of deranged automorphisms and deranged outer automorphisms of a group $H$ will be denoted, respectively, $\operatorname{Aut}_{\mathsf{d}}(H)$ and $\operatorname{Out}_{\mathsf{d}}(H)$.

Consequently, for any finitely generated group $H$ we have

and then,

Note that in (4) we wrote inclusion and not an equality because, in principle, it could happen that a finitely presented $!\operatorname{-by-}\mathbb{Z}$ group has his unique normal subgroup with quotient isomorphic to $\mathbb{Z}$ being not finitely generated. In the next section we shall construct such a group (see LABEL:cor:K*Z_isfg-by-Z_iff_K=1) showing that this inclusion is strict.

Observe that if $H$ is finitely generated or finitely presented, then so is $H\rtimes_{\alpha}\mathbb{Z}$ for every $\alpha\in\operatorname{Aut}(H)$, i.e.,

However, it is less obvious that the converse is not true in general: a semidirect product $H\rtimes_{\alpha}\mathbb{Z}$ can be finitely presented, with $H$ not being finitely generated. Or, as was hinted few lines above, even worse: there do exist finitely presented $\mathbb{Z}$-extensions which are not $\mathbb{Z}$-extensions of any finitely generated group. In other words, the following inclusion is strict:

Indeed, this can happen even for unique $\mathbb{Z}$-extensions. This fact follows easily from the next lemma, showing that any free product $K*\mathbb{Z}$ has the form of a certain semidirect product.

Next, inspired by a trick initially suggested by Maurice Chiodo, we will prove a stronger result. Not only the family !$\mathsf{fg}\operatorname{-by-}\mathbb{Z}$ is a strict subfamily of $[!\operatorname{-by-}\mathbb{Z}]_{\mathsf{fp}}$, but the membership problem between these two families is undecidable: it is impossible to decide algorithmically whether a given finitely presented unique $\mathbb{Z}$-extension is $\mathsf{fg}\operatorname{-by-}\mathbb{Z}$ or not, i.e., whether its unique base group is finitely generated or not. To see this, we use a classic undecidability result: there is no algorithm which, on input a finite presentation, decides whether the presented group is trivial or not (see, for example, [27]).

Of course, if the membership problem is not decidable within some family $\mathcal{H}$, it is also undecidable within any superfamily of $\mathcal{H}$. So, we immediately get the following consequence.

As stated in the introduction, this is exactly the same as saying that the families (of finite presentations) $\mathsf{fp}\operatorname{-by-}\mathbb{Z}$ and $\mathsf{fg}\operatorname{-by-}\mathbb{Z}$ are not recursive. Note that none of these families is neither Markov nor co-Markov, and thus the two undecidability results in 5.4 are not contained in the classic ones due to Adian–Rabin. Indeed, any finitely presented group is a subgroup of some $\mathsf{fp}\operatorname{-by-}\mathbb{Z}$ (and so, of some $\mathsf{fg}\operatorname{-by-}\mathbb{Z}$) group; therefore the families $\mathsf{fp}\operatorname{-by-}\mathbb{Z}$ and $\mathsf{fg}\operatorname{-by-}\mathbb{Z}$ are not Markov. On the other hand, every $\mathsf{fp}$ group embeds in some $2$-generated simple group (see [27, Corollary 3.10]); since

the families $\mathsf{fp}\operatorname{-by-}\mathbb{Z}$ and $\mathsf{fg}\operatorname{-by-}\mathbb{Z}$ are not co-Markov either.

Since the early 1980’s, in a series of papers by R. Bieri, W. Neumann, and R. Strebel (see [3, 4]), several gradually more general invariants—called Sigma (or BNS) Invariants—have been introduced to deal with finiteness conditions for presentations of groups. Concretely in [4], they present an invariant that characterizes those normal subgroups of a finitely generated group $G$ that are finitely generated and contain the commutator $[G,G]$ of $G$. Over the years, this theory has been reformulated in more geometric terms (for a modern version see the survey [32]). Below, we recall this construction and characterization, and discuss some implications of our undecidability results from Section 5.

For a finitely generated group $G$, consider the real vector space $\operatorname{Hom}(G,\mathbb{R})$ of all homomorphisms $\chi\colon G\to\mathbb{R}$ (from $G$ to the additive group of the field of real numbers), which we call characters of $G$. Note that, since $\mathbb{R}$ is abelian and torsion-free, any character $\chi$ must factor through $\pi^{\operatorname{ab}}_{*}$ (abelianizing and then killing the torsion), i.e.,

Thus, $\operatorname{Hom}(G,\mathbb{R})=\operatorname{Hom}(\mathbb{Z}^{r},\mathbb{R})=\mathbb{R}^{r}$, where $r=\operatorname{b}(G)$. We will consider the set of nontrivial characters modulo the equivalence relation given by positive scaling: