When is a finitely generated group finitely presented? I think the question is very general and hard to answer. However I've seen a paper by Baumslag ("Wreath products and finitely presented groups", 1961) showing, as a particular case, that the lamplighter group is not finitely presented. To prove this, he gives conditions to say if a wreath product of groups is finitely presented. The question is: which ways (techniques, invariants, etc) are available to determine whether a finitely generated group is also finitely presented? For instance, is there another way to show that fact about the lamplighter group?
Thanks in advance for references and comments.
 A: One general method is to consider an infinite presentation of the group, and then show that every finite subset of the set of relations defines a group with clearly different property. for example, the lamplighter group has the presentation $\langle \ldots a_{-n}, \ldots, a_1, a_2, \ldots, a_n,\ldots,t \mid a_0^2=1, [a_i,a_j]=1, ta_it^{-1}=a_{i+1}\rangle$. Every finite subpresentation defines a group that has as a quotient one of the following groups $H_n=\langle a_{-n}, \ldots, a_1, a_2, \ldots, a_n,t \mid a_0^2=1, [a_i,a_j]=1, ta_it^{-1}=a_{i+1}\rangle$ for some $n$. The group $H_n$ is an HNN extension of a finite Abelian group $\langle a_{-n},\ldots, a_n\rangle$ with the free letter $t$. Hence $H_n$ is a virtually free group, in particular, $H_n$ contains a non-Abelian free subgroup. Therefore every finite subpresentation defines a group containing a free non-Abelian subgroup, while the Lamplighter group is solvable and thus cannot contain a free non-Abelian subgroup. Similarly   lacunary hyperbolic  but not hyperbolic groups given by presentations satisfying small cancelation conditions or their generalizations are infinitely presented since every finite subpresentation of their presentation defines a hyperbolic group.
I need to add a very nice characterisation of finitely presented metabelian groups: Robert Bieri and Ralph Strebel, Valuations and finitely presented metabelian groups, In: Proceedings of the London Mathematical Society 3.3 (1980), pp. 439-464, https://doi.org/10.1112/plms/s3-41.3.439
A: A seminal work in proving finiteness properties among groups is Brown's article Finiteness properties of groups. The article itself contains applications to Thompson-Higman groups, Houghton's groups and Abels' matrix groups, but since then the strategy has been applied to many other groups.
Another useful strategy comes from Bestvina and Brady's Morse theory. 
The main idea is that, if a group $G$ acts on a (contractible) cellular complex $X$, endowed with a $G$-equivariant Morse function $f : X \to \mathbb{R}$ such that $G$ acts geometrically on the pre-images of bounded intervals under $f$, then there exists a connection between the finiteness properties of $G$ and the contractibility of the ascending and descending links in $X$.
Such a strategy has been applied successfully in many situations, including some subgroups of right-angled Artin groups, Basilica Thompson group, higher dimensional Thompson groups, braided Thompson's groups. (Other applications can be found by looking at the articles citing Bestvina and Brady's paper.)
A: An often-used method is to compute $H_2$.  If the group is finitely presentable then $H_2$ is of finite rank with any coefficients.
For instance, you can use this technique to show that if $q:F\to\mathbb{Z}$ is the map from the free group of rank two that sends both generators to one then the fibre product $H\subseteq F\times F$, ie $(q\times q)^{-1}$ of the diagonal, is infinitely presented.
A famous theorem of Bestvina and Brady shows that this doesn't always work: they give a similar example which is infinitely presented but has finite-rank $H_2$.
A related technique shows that this question is indeed `very hard'.  Grunewald showed that the fibre product coming from a surjection $f:F\to Q$ is finitely presented if and only $Q$ is finite.  It follows that you cannot in general tell if a recursively presented group is (in)finitely presented. 
