Group Theory – Difference Between Finitely Presented and Finitely Generated Groups

Question

A group is said to be finitely generated if it can be generated by a finite set of generators.

Question : Is there difference between finitely presented groups and finitely generated groups?

Answer 1

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.