| The main results are classifications of the solutions of quadratic equations in small cancellation quotients of free groups and free products. Paul E. Schupp has obtained similar results for a restricted class of quadratic equations (those without constants). Some of the methods used here were previously employed by Leo P. Comerford and Charles C. Edmunds in their study of algorithmic problems on the solutions of quadratic equations. The principal tools are the construction of cancellation diagrams on compact surfaces with boundary and an analysis of the co-initial graphs of quadratic words.;Let R be a symmetrized subset of H, and let N be the normal closure of R in H. Let G = H/N and let (phi):H (--->) G be the natural map, so that words of H define elements of G. We look at solutions of W = 1 in G. The tuple (a(,1),...,a(,n)) of words of H is a solution of the equation W((alpha)(,1),...,(alpha)(,n); h(,1),...,h(,k)) in G if the word W(a(,1),...,a(,n); h(,1),...,h(,k)) of H defines the identity element of G.;Much is known about the nature of solutions of quadratic equations in free groups. A free solution of W = 1 in G is induced by a solution of the same equation in H. The solution (a(,1),...,a(,n)) is free if there are words (z(,1),...,z(,n)) of H such that (phi)(z(,i)) = (phi)(a(,i)) for all i and W(z(,1),...,z(,n); h(,1),...,h(,k)) = 1 in H. If we can show that all solutions of W = 1 in a given group G are free, then we can use the classification of solutions in the free group to classify the solutions in G.;If R satisfies sufficiently strong small cancellation conditions, then all solutions of W = 1 in G are free. We first show that a non-free solution induces a reduced R-diagram on a compact surface with boundary. Then we use a combinatorial argument to show that the small cancellation conditions preclude the existence of such a diagram. For this argument we need a lower bound on the Euler characteristics of the surfaces involved; this is where we employ the co-initial graph.;Let F = and H = be free groups. The generators of F and their inverses are called variables, while the generators of H and their inverses are constants. A word W of F*H is quadratic if each variable which occurs in W occurs exactly twice, with exponent +1 or -1 in each occurrence. The words (alpha)(,1)h(,1)(alpha)(,2)(alpha)(,1)('-1)(alpha)(,2)('-1), (alpha)(,1)('2)(alpha)(,2)('2)(alpha)(,3)('2), and (alpha)(,1)h(,1)(alpha)(,2)h(,2)(alpha)(,1)h(,1)(alpha)(,2) are quadratic. If (alpha)(,1),...,(alpha)(,n) and h(,1),...,h(,k) are the variables and constants, respectively, occurring in W, we write W((alpha)(,1),...,(alpha)(,n); h(,1),...,h(,k)).;We also obtain similar results when H is a free product. The constants in the equations are arbitrary nontrivial elements of free factors. |
