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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?

Matthew Kahle
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    The Baumslag–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-Álvarez Jan 25 '12 at 20:17
  • Terry Tao's blogpost http://terrytao.wordpress.com/2008/10/06/finite-subsets-of-groups-with-no-finite-models/ gives a fairly simple example. – David E Speyer Jan 26 '12 at 00:19
  • 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, http://math.stackexchange.com/questions/79852/does-g-cong-g-h-imply-that-h-is-trivial/79907#79907 – user1729 Jan 26 '12 at 10:34
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    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

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