From Notices of the AMS
Some Arithmetic Properties of Complex Local Systems
by Hélène Esnault
Communicated by Han-Bom Moon
Le niveau uniforme du varech sur toutes les roches marquait la ligne de flottaison de la marée pleine et de la mer étale.
(The uniform level of kelp on all the rocks marked the waterline of full tide and slack sea.)
---Victor Hugo, Les travailleurs de la mer, 1866, p.257
Introduction
A group $\pi$ is said to be finitely generated if it is spanned by finitely many letters, that is, if it is the quotient $F\to \pi$ of a free group $F$ on finitely many letters. It is said to be finitely presented if the kernel of such a quotient is itself finitely generated. This does not depend on the choice of generation chosen. For example the trivial group $\pi=\{1\}$ is surely finitely presented as the quotient of the free group in $1$ generator by itself (!). The following finitely presented group shall play a role in the note:
Example 1.
The group $\Gamma_0$ is generated by two elements $(a,b)$ with one relation $b^2=a^2ba^{-2}$.
There are groups which are finitely generated but not finitely presented, see the interesting MathOverflow elementary discussion on the topic .
- Also in Notices
- Presbyopia Correction, Differential Geometry, and Free Boundary PDEs
- Differentiating by Prime Numbers
