On the solvability of a class of equations over groups

Abstract

Let $G$ be a group, $g$ an element of $G$ and $n$ a natural number, $n\geqslant 2$. In [8] B.H. Neumann considered the equation $r_0(t)\deq t^ng=1$ and proved its solvablility in an overgroup. A natural generalisation of this equation is the equation $r(t)=1$, where $r(t)$ is defined as follows: let $W\in G\ast\langle t\rangle$ be a cyclically reduced word of length at least 2 which is not a proper power in $G\ast\langle t\rangle$. Define ${(\ast)}\hspace{4.8cm}r(t)\deq r_0(W)=W^ng$. In this paper we show that for $n\geqslant 3$ the equation $r(t)=1$ is solvable in an overgroup with some structural properties, under certain conditions on $g$ and $W$. It is conjectured in [3, p. 131] that this equation is solvable over an arbitrary group, without conditions on $W$ or $g$.