Sofic approximation sequences and sofic mean dimension

The main purpose of this paper is to strengthen our understanding of sofic mean dimension of two typical classes of sofic group actions. First, we study finite group actions. We prove that sofic mean dimension of any amenable group action does not depend on the choice of sofic approximation sequences. Previously, this result was known only if the acting group is an infinite amenable group. Moreover, we investigate the full shifts, for all sofic groups and all alphabets. We show that sofic mean dimension of any full shift depends purely on its alphabet. Our method is a refinement of the classical technique in relation to the estimates from above and below, respectively, for mean dimension of some typical actions. The key point of our results is that they apply to all compact metrizable spaces without any restriction (in particular, the alphabet concerned in a full shift and the space involved in a finite group action are not required to be finite-dimensional). Furthermore, we improve the quantitative knowledge of sofic mean dimension, restricted to finite-dimensional compact metrizable spaces, for those two typical classes of sofic group actions. As a direct consequence of the main ingredient of our proof, we obtain the exact value of sofic mean dimension of all the actions of finite groups on finite-dimensional compact metrizable spaces. Previously, only an upper bound for these actions was given. Besides, we also get the exact value of sofic mean dimension of full shifts when the alphabet is finite-dimensional.