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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.)

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I must say, that I don't really understand your topic. For example, 2.) is true for any finite group with n large enough, and 3.) and 4.) are not possible, I believe. Also, I don't see how 1.)-5.) are related in any way. –  Kofi Jan 9 '12 at 9:13
@Kofi, regarding 3 and 4, you need to look up the work of Adyan and others on the Burnside problem. I am no specialist in geometric group theory, but I have spent time with people who are, and 1 to 5 have come up as (counter)examples. –  Yemon Choi Jan 9 '12 at 9:34
You should probably put finitely generated infinite groups since it is then the questions become interesting. Also add the ask Mark Sapir tag since he is on MO :). Look at Grigorchuk's survey articles on automaton groups since they have some monsters. –  Benjamin Steinberg Jan 9 '12 at 13:44
There was a workshop on Infinite Monster Groups at Vienna last december: mat.univie.ac.at/%7Earjantseva/monster/programme.html A look at the program might give you ideas about additional properties, and the trends in the subject (which should recall us what the New York Times wrote about Tom Lehrer: ``Mr. Lehrer's muse is not fettered by such inhibiting factors as taste.'') –  Alain Valette Jan 9 '12 at 15:29
Finite group theory has no monsters except the simple groups called the monster and the baby monster, which I think of as cute cuddly monsters. Burnside and others at the turn of the last century naively supposed that finitely generated infinite groups might be nicely behaved as well and proved that for linear groups this is true to some extent. Infinite monster groups are Boogie monsters to many mathematicians because they show that groups can be as wild and uncontrolled as, say, semigroups (even if one must work harder to show it!) if one does not impose extra adjectives like Lie or finite. –  Benjamin Steinberg Jan 10 '12 at 13:42
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3 Answers

(3) and (4) - Tarski Monsters.

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

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I believe Tarski monsters work for 1-4. –  Benjamin Steinberg Jan 9 '12 at 13:49
Yes, you are right! Thanks. –  Steve D Jan 9 '12 at 13:50
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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.

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Caveat: You do not actually want to read Ol'Shanskii's book in detail. It's proofs are often structured very poorly, it contains lots and lots of typos and other minor errors that do not really invalidate anything but give you a really hard time reading the book thoroughly. –  Johannes Hahn Jan 9 '12 at 21:15
@Johannes: If there is a better source for the material in question, I'd love it if you posted it! –  Jon Bannon Jan 10 '12 at 2:40
Olshanskii's book is difficult to read because the material is difficult. But I do not know any significant misprints there, and the construction of proofs is nearly optimal. Of course now there are more conceptually easy methods of constructing extreme groups using the concept of G-subgroup (from more recent papers by Gromov and Olshanskii). See also the paper about lacunary hyperbolic groups. Also rotating families of subgroups developed by Remi Coulon (after Gromov and Delzant) make many constructions conceptually more transparent. –  Mark Sapir Jan 10 '12 at 4:17
Thanks, Mark. This is more valuable than my answer! –  Jon Bannon Jan 10 '12 at 15:23
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You probably should specify "countably infinite".

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

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

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For 2 and maybe 5 he wants f.g. Even if he didnt say it. –  Benjamin Steinberg Jan 9 '12 at 13:48
Groups such that every elements has roots of all orders and are f.g.? I'd really like to see one. –  Maurizio Monge Jan 9 '12 at 19:36
@Maurizio Monge: It think that's exactly the reason for the request for "natural" examples. These monster groups exist and one can often effectively write down presentations, but one does not get a real feeling for these groups. The presentations Ol'shankii gives in his book for the groups with properties 1.-4. (Tarski Monsters) are effective but really, really ugly. For example each relator is (way) more then 10^75 letters long. –  Johannes Hahn Jan 9 '12 at 21:27
Perhaps their presentation is not the best way to understand them, in any case. –  Maurizio Monge Jan 9 '12 at 22:20
@Johannes Hahn: Sorry for not being pedagogically correct, but your comment above is silly. The presentation of any known Tarsky monster is infinite. No wonder most relations are more than $10^{75}$ letters long. On the other hand there are also Tarski monsters satisfying short relations such as $ab=c$. The relations of Olshanskii's "monsters" are actually quite easy to describe. Otherwise the proof would be much longer. –  Mark Sapir Jan 10 '12 at 4:26
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