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1. Entropy, products, and bounded orbit equivalence

   https://doi.org/10.1017/etds.2021.154  

   KERR, DAVID; LI, HANFENG (March 2023, Ergodic Theory and Dynamical Systems)

   Abstract We prove that if two topologically free and entropy regular actions of countable sofic groups on compact metrizable spaces are continuously orbit equivalent, and each group either (i) contains a w-normal amenable subgroup which is neither locally finite nor virtually cyclic, or (ii) is a non-locally-finite product of two infinite groups, then the actions have the same sofic topological entropy. This fact is then used to show that if two free uniquely ergodic and entropy regular probability-measure-preserving actions of such groups are boundedly orbit equivalent then the actions have the same sofic measure entropy. Our arguments are based on a relativization of property SC to sofic approximations and yield more general entropy inequalities. 

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2. Approximate homomorphisms and sofic approximations of orbit equivalence relations

   https://doi.org/10.1017/etds.2024.22  

   HAYES, BEN; KUNNAWALKAM_ELAYAVALLI, SRIVATSAV (December 2024, Ergodic Theory and Dynamical Systems)

   Abstract We show that for every countable group, any sequence of approximate homomorphisms with values in permutations can be realized as the restriction of a sofic approximation of an orbit equivalence relation. Moreover, this orbit equivalence relation is uniquely determined by the invariant random subgroup of the approximate homomorphisms. We record applications of this result to recover various known stability and conjugacy characterizations for almost homomorphisms of amenable groups. 

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3. Equilibrium and nonequilibrium Gibbs states on sofic groups

   https://doi.org/10.1088/1361-6544/adb236  

   Shriver, Christopher (February 2025, Nonlinearity)

   Abstract Recent work of Barbieri and Meyerovitch has shown that, for very general spin systems indexed by sofic groups, equilibrium (i.e. pressure-maximizing) states are Gibbs. The main goal of this paper is to show that the converse fails in an interesting way: for the Ising model on a free group, the free-boundary state typically fails to be equilibrium as long as it is not theonlyGibbs state. For every temperature between the uniqueness and reconstruction thresholds a typical sofic approximation gives this state finite but non-maximal pressure, and for every lower temperature the pressure is non-maximal overeverysofic approximation. We also show that, for more general interactions on sofic groups, the local on average limit of Gibbs states over a sofic approximation Σ, if it exists, is a mixture of Σ-equilibrium states. We use this to show that the plus- and minus-boundary-condition Ising states are Σ-equilibrium if Σ is any sofic approximation to a free group. Combined with a result of Dembo and Montanari, this implies that these states have the same entropy over every sofic approximation. 

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4. A topological dynamical system with two different positive sofic entropies

   https://doi.org/10.1090/btran/101  

   Airey, Dylan; Bowen, Lewis; Lin, Yuqing (February 2022, Transactions of the American Mathematical Society, Series B)

   A sofic approximation to a countable group is a sequence of partial actions on finite sets that asymptotically approximates the action of the group on itself by left-translations. A group is sofic if it admits a sofic approximation. Sofic entropy theory is a generalization of classical entropy theory in dynamics to actions by sofic groups. However, the sofic entropy of an action may depend on a choice of sofic approximation. All previously known examples showing this dependence rely on degenerate behavior. This paper exhibits an explicit example of a mixing subshift of finite type with two different positive sofic entropies. The example is inspired by statistical physics literature on 2-colorings of random hyper-graphs. 

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5. Norm rigidity for arithmetic and profinite groups

   https://doi.org/10.1112/jlms.12719  

   Polterovich, Leonid; Shalom, Yehuda; Shem‐Tov, Zvi (April 2023, Journal of the London Mathematical Society)

   Abstract Let be a commutative ring, and assume that every non‐trivial ideal of has finite index. We show that if has bounded elementary generation then every conjugation‐invariant norm on it is either discrete or precompact. If is any group satisfying this dichotomy, we say that has the dichotomy property . We relate the dichotomy property, as well as some natural variants of it, to other rigidity results in the theory of arithmetic and profinite groups such as the celebrated normal subgroup theorem of Margulis and the seminal work of Nikolov and Segal. As a consequence we derive constraints to the possible approximations of certain non‐residually finite central extensions of arithmetic groups, which we hope might have further applications in the study of sofic groups. In the last section we provide several open problems for further research. 

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