Examples of "Monster" groups - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T15:50:50Z http://mathoverflow.net/feeds/question/85236 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/85236/examples-of-monster-groups Examples of "Monster" groups Owen Sizemore 2012-01-09T07:38:08Z 2013-03-05T05:08:37Z <p>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:</p> <p>1.) Non-amenable groups without free subgroups</p> <p>2.) Groups such that $x^n=e \hspace10pt\forall x$</p> <p>3.) Groups with all proper subgroups cyclic</p> <p>4.) Groups such that every proper subgroup is finite and cyclic of a given order</p> <p>5.) Groups such that every elements has roots of all orders</p> <p>The main source I have been using so far is a survey by Mark Sapir <a href="http://arxiv.org/abs/0704.2899" rel="nofollow">http://arxiv.org/abs/0704.2899</a>.</p> <p>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.) </p> http://mathoverflow.net/questions/85236/examples-of-monster-groups/85237#85237 Answer by Ricky Demer for Examples of "Monster" groups Ricky Demer 2012-01-09T08:02:57Z 2012-01-09T08:02:57Z <p>You probably should specify "countably infinite".</p> <p>${}$2. $\qquad$ $\left\langle\displaystyle\bigoplus_{j=0}^{\infty} \;\; \mathbb{Z}/n\mathbb{Z} \;\;\; , \;\;\; + \right\rangle$</p> <p>${}$5. $\qquad$ $\langle\mathbb{Q},+\rangle$</p> http://mathoverflow.net/questions/85236/examples-of-monster-groups/85256#85256 Answer by Steve D for Examples of "Monster" groups Steve D 2012-01-09T13:39:03Z 2012-01-09T13:50:20Z <p>(3) and (4) - Tarski Monsters.</p> <p>EDIT - Benjamin Steinberg pointed out this works for (1) and (2) as well.</p> http://mathoverflow.net/questions/85236/examples-of-monster-groups/85284#85284 Answer by Jon Bannon for Examples of "Monster" groups Jon Bannon 2012-01-09T20:02:54Z 2012-01-09T20:02:54Z <p>Hi Owen,</p> <p>This is really a long comment, but points to a number of things that answer this question.</p> <p>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 <a href="http://front.math.ucdavis.edu/0701.5365" rel="nofollow">lacunary hyperbolic groups</a>. Additionally, have a look at Olshanskii's book <a href="http://www.amazon.com/Geometry-Defining-Relations-Mathematics-Applications/dp/0792313941" rel="nofollow">Geometry of defining relations in groups</a>. In this book many such examples are generated that solve various problems. </p>