A Conjecture about Conjugacy in Free Groups
نویسندگان
چکیده
Say that an element of a free group is a pure conjugate if it can be expressed from the generators using exclusively the conjugacy operation. We study free reductions in words representing pure conjugates. Using finite state automata, we attribute to the letters in such words levels that live in some free left distributive system. If a certain conjecture about this system is true, then reduction can occur only between letters lying on the same level. Under this conjecture, we establish restrictions on the form of those identities satisfied by group conjugacy, and we construct unique normal forms for large families of pure conjugates. We also show how to use group conjugacy to solve a problem related to the word problem of left self-distributivity.
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