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Abstract: Let $F= < a,b>$ be a rank two free group. A word $Wa,b$ in $F$ is {\slprimitive} if it, along with another group element, generates the group. It isa {\sl palindrome} with respect to $a$ and $b$ if it reads the same forwardsand backwards. It is known that in a rank two free group any primitive elementis conjugate either to a palindrome or to the product of two palindromes, butknown iteration schemes for all primitive words give only a representative forthe conjugacy class. Here we derive a new iteration scheme that gives eitherthe unique palindrome in the conjugacy class or expresses the word as a uniqueproduct of two unique palindromes. We denote these words by $E {p-q}$ where$p-q$ is rational number expressed in lowest terms. We prove that $E {p-q}$ isa palindrome if $pq$ is even and the unique product of two unique palindromesif $pq$ is odd. We prove that the pairs $E {p-q},E {r-s}$ generate the groupwhen $|ps-rq|=1$. This improves the previously known result that held only for$pq$ and $rs$ both even. The derivation of the enumeration scheme also gives anew proof of the known results about primitives.



Autor: Jane Gilman, Linda Keen

Fuente: https://arxiv.org/







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