## Finitely presented groups which are not residually amenable

### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

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!

-

@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

## 3 Answers

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.

-

thanks for your example – Kate Juschenko Jun 15 2011 at 19:35

### You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.

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.)

-

thanks for the citation and example. – Kate Juschenko Jun 15 2011 at 19:35
You're welcome. – Richard Kent Jun 15 2011 at 21:07

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

-

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
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 15, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9170600771903992, "perplexity_flag": "middle"}