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What are examples of finitely presented but not residually amenable groups?

Well, the examples that I want to have are simple f.p. groups as well as examples of non residually amenable groups arise from other reasoning then simplicity.

Thank you for all your references!

edited Jun 13 2011 at 16:13

Kate Juschenko

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	@Kate: Take a non-residually-finite group $G$ with property $T$. Then $G$ cannot be residually-amenable (since every amenable quotient of $G$ is finite). Examples of fp groups like this are given some some lattices in non-linear Lie groups, see e.g., J.Millson, "Real vector bundles with discrete structure group", Topology 18 (1979) 83–89, and M.Raghunathan, "Torsion in co-compact lattices of Spin(2,N)", Math. Ann. 266 (1984) 403–419. None of these groups is simple (they also do not contain infinite simple subgroups). This answers your 2nd question. – Misha Apr 13 2012 at 4:56
	Thank you, Misha, for your answer and the citation! – Kate Juschenko Apr 24 2012 at 7:41

Guntram · Answer 1 · 2011-06-13 12:34:17Z UTC

Let $G$ be an adjoint Kac-Moody group over a (sufficiently large) finite field $\mathbf F_q$. By results of Caprace-R'emy, $G$ is simple when its diagram is connected and has indefinite type, i.e. neither spherical nor affine, and finitely presented when the diagram does not contain an edge labelled with $\infty$. In this case, $G$ itself is not amenable as it contains the free product of two root groups $U_\alpha * U_\beta$.

Varying the ground field and the diagram then gives a two-parameter family of examples.

Richard Kent · Answer 2 · 2011-06-13 12:59:05Z UTC

Take any finitely presented infinite simple group $G$. It is not residually anything (well, it is residually $G$).

Now take such a $G$ that contains a nonabelian free group. For example, take Elizabeth Scott's finitely presented group $G$ that contains $GL_3(\mathbb{Z})$. (See Scott, Elizabeth A. The embedding of certain linear and abelian groups in finitely presented simple groups. J. Algebra 90 (1984), no. 2, 323–332.)

Alain Valette · Answer 3 · 2011-06-13 20:28:31Z UTC

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Cornulier has a finitely presented sofic group which is not the limit of amenable groups: http://arxiv.org/pdf/0906.3374

answered Jun 13 2011 at 20:28

Alain Valette

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	Thanks, Alain, in fact this paper was my starting point... – Kate Juschenko Jun 14 2011 at 12:05
	Also, I've received several very good references form Mark Sapir on finitely presented simple groups. The question I wanted to ask is that there could be much more examples of finitely presented (non-simple) groups that are not residually amenable. – Kate Juschenko Jun 14 2011 at 14:25

Finitely presented groups which are not residually amenable

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