  In this article we study the automorphism group ${\rm Aut}(X,\sigma)$ of
subshifts $(X,\sigma)$ of low word complexity. In particular, we prove that
Aut$(X,\sigma)$ is virtually $\mathbb{Z}$ for aperiodic minimal subshifts and
certain transitive subshifts with non-superlinear complexity. More precisely,
the quotient of this group relative to the one generated by the shift map is a
finite group. In addition, we show that any finite group can be obtained in
this way. The class considered includes minimal subshifts induced by
substitutions, linearly recurrent subshifts and even some subshifts which
simultaneously exhibit non-superlinear and superpolynomial complexity along
different subsequences. The main technique in this article relies on the study
of classical relations among points used in topological dynamics, in
particular, asymptotic pairs. Various examples that illustrate the technique
developed in this article are provided. In particular, we prove that the group
of automorphisms of a $d$-step nilsystem is nilpotent of order $d$ and from
there we produce minimal subshifts of arbitrarily large polynomial complexity
whose automorphism groups are also virtually $\mathbb{Z}$.
