Solutions of quadratic equations in groups with small cancellation condition (Q1262952)

The author studies quadratic equations over small cancellation groups. He considers a group G which has a finite presentation \(<A\); \(r=1\) \((r\in R)>\) satisfying one of the small cancellation conditions C(7), C(5) \& T(4), C(4) \& T(5), C(3) \& T(7). Let H and F be free groups freely generated, respectively, by A and X disjoint from A. By an equation over G the author means an expression \(W=1\), where W is an element of F*H the free product of F and H. An equation \(W=1\) is called quadratic if each element of X which occurs in W occurs exactly twice, each time with the exponent \(+1\) or -1. The author says that an equation \(\phi (x_ 1,...,x_ n)=1\) over G reduces to the system of equations \(\phi_ i(x_ 1,...,x_ n,y_ 1,...,y_ k)=1\) (i\(\in I)\) over a free group if \((u_ 1,...,u_ n)\) is a solution to the equation \(\phi =1\) over G if and only if there exists a solution \((u'_ 1,...,u'_ n,w_ 1,...,w_ k)\) of some equation \(\phi_ i=1\) over the free group freely generated by A such that \(u_ i=^{G}u'_ i\) for all \(1\leq i\leq n\). The author proves that if \(\phi\) is a quadratic equation over the considered group G, \(N=80| \phi |\), \(\{y_ 1,...,y_ N\}\) is a set of variables disjoint from \(X^{\pm 1}\cup A^{\pm 1}\), then the equation \(\phi =1\) over G reduces to the system of equations \(\phi y_ 1^{-1}r_ 1y_ 1...y_ v^{-1}r_ vy_ v=1\) (0\(\leq v\leq N\), \(r_ 1,...,r_ v\in R)\) over a free group.