I am looking for books, papers, or even webpages, that have collected many counter examples in group theory (which, I guess, are just examples in group theory).
I am particularly interested in sources with a focus on infinite, finitely generated, groups. As some examples, a reference may contain things like:
Tarski monster groups: an infinite groups where every element has the same finite order, non-amenable group that does not contain a free subgroup of rank two etc
Baumslag-Solitar groups give examples of finitely presented (one-relator) non-Hopfian groups, etc
Groups of intermediate growth, they are non-elementarily amenable and they have intermediate growth.
Infinite groups without a non-discrete Hausdorff group topology
Fintely presented groups with strange properties (where maybe infinitely presented are easier to find/construct)
Finitely presented groups without solvable word problem.
Counter examples restricted to groups that are considered nicer than arbitrary finitely generated group (finitely presented groups), like hyperbolic groups or automatic groups.
For online references it would be especially cool if you could search for groups with certain properties in a similar vain to the $\pi$-base website. Although I would really like such a site to contain references to the literature. (I don't think such a site exists, as I would have found it but, hopefully I am mistaken)
Please one source per answer, and if possible summarize what it covers. I know it it is basically hopeless for one central source, so I made this a big list. Plus this opens it up to references that are not focused on infinite, finitely presented sources which should hopefully make this more useful to others.
A sort of relevent question, on mathoverflow, is Counterexamples in Algebra?