# What is the easiest example of a finitely presented group which is not residually finite?

What is an easy example of a finitely presented group which is not residually finite?

To be clear, part of the question is: how do we see that it isn't?

-
The Baumslagâ€“Solitar group B(2,3) is a classical example, it has an incredibly simple presentation, and showing it is not hopfian (and therefore non-residually finite) is done using the basic tools of combinatorial group theory... What does easy mean, exactly? – Mariano Suárez-Alvarez Jan 25 '12 at 20:17
Terry Tao's blogpost terrytao.wordpress.com/2008/10/06/… gives a fairly simple example. – David Speyer Jan 26 '12 at 0:19
Further to MarianoSuárez-Alvarez's comment, there is a classification of Baumslag-Solitar groups with respect to whether they are Hopfian, residually finite, or neither. See my post here, math.stackexchange.com/questions/79852/… – user1729 Jan 26 '12 at 10:34
Also, there is a recent paper (2007) by, among others, the Baumslag of Baumslag-Solitar fame, called "Reflections on the residual finiteness of one-relator groups". In this paper the authors prove that if $r$ and $w$ are non-commuting words in the free group $F(a, b, \ldots)$ then the group $\langle a, b, \ldots ; r^{r^w}=r^2\rangle$ is not residually finite. Which is pretty. – user1729 Jan 26 '12 at 10:49