Finitely presented groups which are not residually amenable - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T09:32:13Z http://mathoverflow.net/feeds/question/67645 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/67645/finitely-presented-groups-which-are-not-residually-amenable Finitely presented groups which are not residually amenable Kate Juschenko 2011-06-13T11:53:06Z 2011-06-13T20:28:31Z <p>What are examples of finitely presented but not residually amenable groups?</p> <p>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.</p> <p>Thank you for all your references!</p> http://mathoverflow.net/questions/67645/finitely-presented-groups-which-are-not-residually-amenable/67651#67651 Answer by Guntram for Finitely presented groups which are not residually amenable Guntram 2011-06-13T12:34:17Z 2011-06-13T12:34:17Z <p>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 $U_\alpha * U_\beta$. </p> <p>Varying the ground field and the diagram then gives a two-parameter family of examples.</p> http://mathoverflow.net/questions/67645/finitely-presented-groups-which-are-not-residually-amenable/67654#67654 Answer by Richard Kent for Finitely presented groups which are not residually amenable Richard Kent 2011-06-13T12:59:05Z 2011-06-13T12:59:05Z <p>Take any finitely presented infinite simple group $G$. It is not residually anything (well, it is residually $G$).</p> <p>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.) </p> http://mathoverflow.net/questions/67645/finitely-presented-groups-which-are-not-residually-amenable/67702#67702 Answer by Alain Valette for Finitely presented groups which are not residually amenable Alain Valette 2011-06-13T20:28:31Z 2011-06-13T20:28:31Z <p>Cornulier has a finitely presented sofic group which is not the limit of amenable groups: <a href="http://arxiv.org/pdf/0906.3374" rel="nofollow">http://arxiv.org/pdf/0906.3374</a></p>