Word equations in a uniquely divisible group
Abstract: We study equations in groups G with unique m-th roots for each
integer m. A word equation in two letters is an expression of the form
w(X,A) = B, where w is a finite word in the alphabet {X,A}. We think of
A,B in G as fixed coefficients, and X in G as the unknown. Certain word
equations, such as XAXAX=B, have solutions in terms of radicals: X =
A^{-1/2}(A^{1/2}BA^{1/2})^{1/3}A^{-1/2}, while others such as XAX^2 = B do
not. We obtain the first known infinite families of word equations not
solvable by radicals, and conjecture a complete classification, which can
be thought of as a noncommutative Abel theorem.
One of our main tools is a new combinatorial gadget: Given a word w we
associate to it a word polynomial Pw in Z[x,y] in two commuting variables,
which factors whenever w is a composition of smaller words. Our main
result is then that if the equation Pw(x^2,y^2)=0 has a nontrivial
solution mod p for all but finitely many primes p, then the word equation
w(X,A)=B is not solvable in terms of radicals. We apply this theorem along
with some known results on equations over finite fields to produce
infinite families of word equations not solvable by radicals. Finally, we
shall discuss connections of our work to the long-standing BMV trace
conjecture in statistical physics, which gave us inspiration for the
questions we study here. (Joint work with Lionel Levine and Darren Rhea).