Twisted conjugacy classes in special and general linear groups

In this paper we study twisted conjugacy classes and the $R_{\mathrm{\infty }}$-property for classical linear groups. In particular, we prove that the general linear group ${\mathrm{G}\mathrm{L}}_n(K)$ and the special linear group ${\rm SL}n(K)$possess$R{\infty}$-property,if$n\geq3$and$K$ is an infinite integral domain with trivial group of automorphisms (Theorem 1), or $K$ is an integral domain, which has a zero characteristic and for which $\mathrm{A}\mathrm{u}\mathrm{t}(K)$ is torsion (Theorem 2).