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.)