TITLE: Reference about the Banach-Tarski paradox
QUESTION [9 upvotes]: I think the title says it all. I am planning on giving a talk in a few weeks about the Banach-Tarski paradox and I have some pdfs found online which describe the paradox a little but I am looking for a solid reference which covers the construction from A to Z and on which I can extract the main ideas for my talk from (I understand the ideas beneath the paradox, I am just looking for a formal proof with no details excluded,i.e. a well-structured document). Anyone has a reference in mind?

REPLY [13 votes]: I gave a similar presentation at MathFest 2011 in Kentucky last week, using Stan Wagon's book as a guide.  Here is a list of definitions/theorems/etc that are on the direct thread to reaching the Banach-Tarski Paradox (stated as a corollary in the book).

Def 1.1: G-Paradoxical
Thm 1.2: Free group of Rank 2 F-paradoxical
Prop 1.10: Group acting without nontrivial fixed points
Thm 2.1: $SO_3$ has free subgroup of Rank 2 
Thm 2.3: Hausdorff Paradox 
Def 3.3: G-Equidecomposable 
Prop 3.4: Equidecomposability preserves Paradoxes 
Thm 3.9: $S^2$ and $S^2$ minus a countable set are equidecomposable 
Cor 3.10: The Banach-Tarski Paradox

As others have stated, there are MANY interesting results along the way in this book, and the development is superb.  Here is a link to a modified version of the presentation I gave at MathFest.  It is an attempt at illustrating exactly what is presented in the material of the text, rather than providing an alternative interpretation (baby steps, right?).  For the web version, I've added some annotations so that it's better suited for reading as the original slide presentation didn't have a lot of textual development, though unfortunately I was not able to add detailed descriptions of the animations without significant re-work.