Examples of "Monster" groups - MathOverflow

Examples of "Monster" groups

2012-01-09T07:38:08Z

I am planning a talk for a general graduate student audience. The topic is exotic examples of countable discrete groups ("monsters"). Some examples of properties that I'm interested in are:

1.) Non-amenable groups without free subgroups

2.) Groups such that $x^n=e \hspace10pt\forall x$

3.) Groups with all proper subgroups cyclic

4.) Groups such that every proper subgroup is finite and cyclic of a given order

5.) Groups such that every elements has roots of all orders

The main source I have been using so far is a survey by Mark Sapir http://arxiv.org/abs/0704.2899.

I would like additional sources. Additional properties to the ones above would also be great. Also examples that arise "naturally" (say as a group of symmetries of some nice space rather than a combinatorial construction would be great.)

Answer by Ricky Demer for Examples of "Monster" groups

2012-01-09T08:02:57Z

You probably should specify "countably infinite".

${}$2. $\qquad$ $\left\langle\displaystyle\bigoplus_{j=0}^{\infty} \;\; \mathbb{Z}/n\mathbb{Z} \;\;\; , \;\;\; + \right\rangle$

${}$5. $\qquad$ $\langle\mathbb{Q},+\rangle$

Answer by Steve D for Examples of "Monster" groups

2012-01-09T13:39:03Z

(3) and (4) - Tarski Monsters.

EDIT - Benjamin Steinberg pointed out this works for (1) and (2) as well.

Answer by Jon Bannon for Examples of "Monster" groups

2012-01-09T20:02:54Z

Hi Owen,

This is really a long comment, but points to a number of things that answer this question.

Mark Sapir has made some useful comments in answers to a couple of my questions that may be helpful to you. One that comes to mind is his mention of lacunary hyperbolic groups. Additionally, have a look at Olshanskii's book Geometry of defining relations in groups. In this book many such examples are generated that solve various problems.