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→Paradoxical Decomposition of the Free Group
[[Image:Free group decomp.jpg|right|200px|thumb|U,L, and D are within the ovals. R is what is left over.]]
[[Image:Free group decomp.jpg|right|200px|thumb|U,L, and D are within the ovals. R is what is left over.]]
Why are we talking about the free group? It is important for two reasons, the first of which is that it admits a "paradoxical decomposition", if we allow such a notion to refer to the graphical representation of it above. We begin by partioning the group as <math>F_2 = U \cup D \cup L \cup R \ </math>, where the sets <math>U,D,L,R \ </math> are those described in the image to the right.
Why are we talking about the free group? It is important for two reasons, the first of which is that it admits a "paradoxical decomposition", if we allow such a notion to refer to the graphical representation of it above. We begin by partioning the group as <math>F_2 = U \cup D \cup L \cup R \ </math>, where the sets <math>U,D,L,R \ </math> are those described in the image to the right.
Consider <math>b'(U) = \left\{{b'u : u \in U }\right\} \ </math>. Since every element of ''U'' starts with <math>b^k \ </math>, followed by a letter <math>a, a' \ </math>, and then some other letters, every element of <math>b'(U) \ </math> starts with <math>b, a, \ </math> or <math>a' \ </math> - that is, three of the four trees of <math>F_2 \ </math>! The only thing missing is <math>D \ </math>. Thus, <math>b'(U) \cup D = F_2 \ </math>. Likewise, <math>a'(R) \cup L = F_2 \ </math>. Hence, we have something akin to a "paradoxical decomposition" of the free group.
===A Geometric Model of the Free Group===
===A Geometric Model of the Free Group===