Free products in linear groups

Abstract:

Let $R$ be a commutative integral domain of characteristic $0$, and let $G$ be a finite subgroup of $\mathrm {PGL}_n(R)$, the projective general linear group of degree $n$ over $R$. In this note, we show that if $n\geq 2$, then $\mathrm {PGL}_n(R)$ also contains the free product $G*T$, where $T$ is the infinite cyclic group generated by the image of a suitable transvection.