When is a finitely generated group finitely presented? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T06:42:38Z http://mathoverflow.net/feeds/question/54975 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/54975/when-is-a-finitely-generated-group-finitely-presented When is a finitely generated group finitely presented? Mauricio 2011-02-10T02:10:38Z 2011-04-23T16:27:44Z <p>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?</p> <p>Thanks in advance for references and comments.</p> http://mathoverflow.net/questions/54975/when-is-a-finitely-generated-group-finitely-presented/54978#54978 Answer by Mark Sapir for When is a finitely generated group finitely presented? Mark Sapir 2011-02-10T02:36:59Z 2011-02-10T03:39:11Z <p>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 <a href="http://front.math.ucdavis.edu/0701.5365" rel="nofollow"> lacunary hyperbolic </a> 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. </p> http://mathoverflow.net/questions/54975/when-is-a-finitely-generated-group-finitely-presented/54982#54982 Answer by HW for When is a finitely generated group finitely presented? HW 2011-02-10T03:33:05Z 2011-02-10T03:33:05Z <p>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.</p> <p>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.</p> <p>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 $H_2$.</p> <p>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. </p>