Finitely presented groups which are not residually amenable - MathOverflow

Finitely presented groups which are not residually amenable

2011-06-13T11:53:06Z

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!

Answer by Guntram for Finitely presented groups which are not residually amenable

2011-06-13T12:34:17Z

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.

Answer by Richard Kent for Finitely presented groups which are not residually amenable

2011-06-13T12:59:05Z

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.)

Answer by Alain Valette for Finitely presented groups which are not residually amenable

2011-06-13T20:28:31Z

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