On the probability of satisfying a word in a group

Abstract

We show that for any finite group $G$ and for any $d$ there exists a word $w\in F_{d}$ such that a $d$-tuple in $G$ satisfies $w$ if and only if it generates a solvable subgroup. In particular, if $G$ itself is not solvable, then it cannot be obtained as a quotient of the one relator group $F_{d}/<w>$. As a corollary, the probability that a word is satisfied in a fixed non-solvable group can be made arbitrarily small, answering a question of Alon Amit.