A. J. Berrick has an interesting paper explaining how a topologist thinks about group theory. Topology and group theory are connected throught the fundamental group. For every group, topologists can construct a space with that group as its fundamental group. Some of these can be very complicated, even for comparatively uncomplicated groups. For example, perfect groups lead to very scary-looking constructions.
The paper is A topologist’s view of perfect and acyclic groups.
Your link to the Wikipedia article on perfect groups has a typo.
Fixed. Thanks!
Cool article. So far, what I like about it most is that it gives me hope for really understanding Quillen’s plus construction! On page 6, it talks about a “Kan-Thurston theorem”, which says that the homotopy category of spaces is equivalent to a category where an object is a group together with a perfect normal subgroup.
I had the exact same reaction.