Topological pressure and the variational principle for actions of sofic groups (Q2873989)

The author introduces a definition of topological pressure for countable sofic groups \(\Gamma\) acting on a compact metric space \(X\). This generalizes the definition of topological pressure for amenable groups given by D. Kerr and H. Li. Sofic groups were introduced in different ways by M. Gromov and by B. Weiss. The main feature of sofic groups \(\Gamma\) is the existence of a sequence of integers \(\{d_N \}\) and a sequence of ``quasi-representations'' \(\sigma_N\) of \(\Gamma\) in the group of permutations of \(\{1, 2, \dots, d_N\}\). One idea used is that the action of an element \(\gamma \in \Gamma\) on a \(d_N\)-uple \((x_1, \dots x_{d_N})\) is modelled by the maps \(\sigma_N\) in the sense that \(\gamma x_i\) is close to \(x{}_{\sigma}{}_{N(\gamma )(i)}\), \(i = 1, 2,\dots, d_N\). In sofic groups, averages over Følner sequences can be replaced by averages over the sets \(\{1, 2, \dots, d_N\}\). In this way, she can define the topological pressure introduced in the article under review. The main results proved in this article are that for amenable groups the topological pressure introduced agrees with that defined by Kerr and Li. The other main theorem is a variational principle for the topological pressure of sofic action groups. For this, the author introduces a sofic measure entropy. Then he gives the concept of equilibrium states in the same way as in the classical thermodynamic formalism.