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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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    $\begingroup$ @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. $\endgroup$
    – Yemon Choi
    Jan 9, 2012 at 9:34
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    $\begingroup$ 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. $\endgroup$ Jan 9, 2012 at 13:44
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    $\begingroup$ 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.'') $\endgroup$ Jan 9, 2012 at 15:29
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    $\begingroup$ Re (1): there is a f.p. example constructed by Olshanskii and Sapir, see this answer of Denis Osin mathoverflow.net/questions/78410/… elsewhere on MO $\endgroup$
    – Yemon Choi
    Jan 9, 2012 at 16:08
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    $\begingroup$ 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. $\endgroup$ Jan 10, 2012 at 13:42

2 Answers 2

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(3) and (4) - Tarski Monsters.

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

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  • $\begingroup$ I believe Tarski monsters work for 1-4. $\endgroup$ Jan 9, 2012 at 13:49
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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 reference that comes to mind is his Lacunary hyperbolic groups with Olshanskii and Osin. 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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  • $\begingroup$ 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. $\endgroup$ Jan 9, 2012 at 21:15
  • $\begingroup$ @Johannes: If there is a better source for the material in question, I'd love it if you posted it! $\endgroup$
    – Jon Bannon
    Jan 10, 2012 at 2:40
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    $\begingroup$ 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. $\endgroup$
    – user6976
    Jan 10, 2012 at 4:17

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