Finitely generated group which is not finitely presented [duplicate]

The following group is finitely generated but not finitely presentable: $$ G=\langle a, b, t\mid t[a^i, b^j]t^{-1}=[a^i, b^j], i, j\in\mathbb{Z}\rangle $$ It is clearly finitely generated. To see that it is not finitely presentable, note that it is an HNN-extension whose associated subgroup is free of infinite rank (the associated subgroup is in fact the derived subgroup $F(a, b)'$, which is not finitely generated). This means that the given presentation is aspherical*, and hence minimal. It is then "well known" that such a group $G$ cannot be finitely presented. One reason is as follows: suppose that $H$ is a finitely presentable group, and that $H$ has presentation $\langle \mathbb{x}; \mathbf{r}\rangle$ with $\mathbf{x}$ finite and $\mathbf{r}$ infinite. Then all but finitely many of the relators are redundant: there exists a subset $\mathbf{s}\subset \mathbf{r}$ such that $\mathbf{s}$ is finite and such that $\langle\langle\mathbf{s}\rangle\rangle=\langle\langle\mathbf{r}\rangle\rangle$. In our example, this cannot happen by asphericity/minimality. Hence, $G$ is not finitely presentable.

*Chiswell, I.M., D.J. Collins, and J.Huebschmann. Aspherical group presentations. Math. Z. 178.1 (1981): 1-36.

I'm not sure how much group theory you're willing to assume. Does the following argument answer your question?

The given group $G$ has presentation $$ \langle a,b \mid [a^{-n}ba^n, b] = 1\text{ for }n\in\mathbb{N}\rangle. $$ Consider then the following sequence of groups $$ G_n \;=\; \langle a,b \mid [a^{-1}ba,b] = \cdots = [a^{-n}ba^n,b]=1\rangle $$ These groups fit into a natural sequence of quotient homomorphisms $$ G_1 \to G_2 \to G_3 \to \cdots $$ Then $G$ is finitely presented if and only if this sequence eventually stabilizes.

There are many different ways to show that this sequence does not stabilize. For example, each $G_n$ has a natural homomorphism to $\mathbb{Z}$ which sends $a$ to $1$ and $b$ to $0$. By Schreier's lemma, the kernel $K_n$ of this homomorphism is generated by the elements $b_i = a^{-i}ba^i$ and has presentation $$ K_n = \langle \ldots,b_{-1},b_0,b_1,b_2\ldots \mid [b_i,b_j]=1\text{ for }|i-j|\leq n\rangle. $$ The resulting group clearly depends on $n$. For example, if we fix a value of $m$ and add the relations $b_i = 1$ for $i \in \mathbb{Z}-\{0,m\}$, the resulting quotient of $K_n$ is abelian if $m\leq n$, and is a nonabelian free group of rank two if $m > n$.

Another alternative is to consider the quotient of $G_n$ obtained by adding the relations $a^m = b^2 = 1$. The resulting quotient is finite if and only if $m \leq 2n+1$, so all of the $G_n$'s must be different.

Here is a sketch of proof:

• $\langle a,t \mid [t^nat^{-n},t^{m}at^{-m} ]=1, \ n,m \in \mathbb{Z} \rangle$ is a presentation of $L:= \mathbb{Z} \wr \mathbb{Z}$.

• If $L$ is finitely presented, there exists a finite set $S \subset \mathbb{Z}$ such that $$\langle a,t \mid [t^nat^{-n},t^{m}at^{-m}]=1, \ n,m \in S \rangle$$ is isomorphic to $L$, so it is sufficient to show that $$L_k= \langle a,t \mid [t^nat^{-n},t^{m}at^{-m}]=1, -k \leq n,m \leq k \rangle$$ is not isomorphic to $L$.

• Setting $a_n=t^nat^{-n}$, we get $$L_k= \langle a_{-k}, \dots, a_k,t \mid [a_n,a_{m}]=1, a_{n+1}=ta_nt^{-1}, -k \leq n,m \leq k \rangle.$$

• Notice that $L_k$ is a HNN extension and use Britton lemma to show that the subgroup $\langle a_0,t \rangle$ is isomorphic to the free product $\mathbb{Z} \ast \mathbb{Z}$.

• Therefore, $L_k$ is not solvable so it cannot be isomorphic to $L$ (which is solvable).

Yes. For instance, $(\mathbb{Z},+)$ is finitelty presented (it is generated by $1$), but it is not finite.

On the other hand, every finite group is finitely presented.

Rips' construction gives lots of examples of finitely generated groups which are not finitely presentable. Rips* proved the following result.

Theorem. For every finitely presented group $Q$ there exists a hyperbolic group $H$ and a finitely generated, normal subgroup $N$ of $H$ such that $H/N\cong Q$.

Given a finite presentation of $Q$, Rips explicitly constructs the group $H$. This result is usually referred to as Rips' construction. It turns out that in Rips' construction the subgroup $N$ is finitely presentable if and only if the image group $Q$ is finite (see Exercise II.5.47, p227, of Bridson and Haefliger, Metic spaces of non-positive curvature - one direction is obvious, while the other direction is highly non-trivial).

* E. Rips, Subgroups of small Cancellation Groups, Bulletin of the London Mathematical Society, Volume 14, Issue 1, 1 January 1982, pp45–47, doi link.