  A countable group $G$ has the strong topological Rokhlin property (STRP) if
it admits a continuous action on the Cantor space with a comeager conjugacy
class. We show that having the STRP is a symbolic dynamical property. We prove
that a countable group $G$ has the STRP if and only if certain sofic subshifts
over $G$ are dense in the space of subshifts. A sufficient condition is that
isolated shifts over $G$ are dense in the space of all subshifts.
  We provide numerous applications including the proof that a group that
decomposes as a free product of finite or cyclic groups has the STRP. We show
that finitely generated nilpotent groups do not have the STRP unless they are
virtually cyclic; the same is true for many groups of the form $G_1\times
G_2\times G_3$ where each factor is recursively presented. We show that a large
class of non-finitely generated groups do not have the STRP, this includes any
group with infinitely generated center and the Hall universal locally finite
group.
  We find a very strong connection between the STRP and shadowing, a.k.a.
pseudo-orbit tracing property. We show that shadowing is generic for actions of
a finitely generated group $G$ if and only if $G$ has the STRP.
