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A countable group $G$ has the strong topological Rokhlin property (STRP) ifit admits a continuous action on the Cantor space with a comeager conjugacyclass. We show that having the STRP is a symbolic dynamical property. We provethat a countable group $G$ has the STRP if and only if certain sofic subshiftsover $G$ are dense in the space of subshifts. A sufficient condition is thatisolated shifts over $G$ are dense in the space of all subshifts.  We provide numerous applications including the proof that a group thatdecomposes as a free product of finite or cyclic groups has the STRP. We showthat finitely generated nilpotent groups do not have the STRP unless they arevirtually cyclic as well as 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 ofnon-finitely generated groups do not have the STRP, this includes any groupwith 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 ofa countable group $G$ if and only if $G$ has the STRP.  We also consider the STRP for totally disconnected locally compact groups andcharacterize pro-countable metrizable groups with the STRP. In particular,inverse limits of groups with the STRP have the STRP and so every pro-finitegroup has the STRP.

Strong topological Rokhlin property, shadowing, and symbolic dynamics of countable groups