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We present several applications of the weak specification property andcertain topological Markov properties, recently introduced by S. Barbieri, F.Garc\'{i}a-Ramos and H. Li, and implied by the pseudo-orbit tracing property,for general expansive group actions on compact spaces.  First we show that any expansive action of a countable amenable group on acompact metrizable space satisfying the weak specification and strongtopological Markov properties satisfies the Moore property, i.e. everysurjective automorphism of such dynamical system is pre-injective. Thistogether with an earlier result of H. Li (where the strong topological Markovproperty is not needed) of the Myhill property, which we also re-prove here,establishes the Garden of Eden theorem for all expansive actions of countableamenable groups on compact metrizable spaces satisfying the weak specificationand strong topological Markov properties. We hint how to easily generalize thisresult even for uncountable amenable groups and general compact, notnecessarily metrizable, spaces.  Second, we generalize the recent result of D. B. Cohen that any subshift offinite type of a finitely generated group having at least two ends has weaklyperiodic points. We show that every expansive action of such a group having acertain Markov topological property, again implied by the pseudo-orbit tracingproperty, has a weakly periodic point. If it has additionally the weakspecification property, the set of such points is dense.

Garden of Eden and weakly periodic points for certain expansive actions of groups