Strongly sofic monoids, sofic topological entropy, and surjunctivity

Abstract

We introduce the class of strongly sofic monoids. This class of monoids strictly contains the class of sofic groups and is a proper subclass of the class of sofic monoids. We define and investigate sofic topological entropy for actions of strongly sofic monoids on compact spaces. We show that sofic topological entropy is a topological conjugacy invariant for such actions and use this fact to prove that every strongly sofic monoid is surjunctive. This means that if $M$ is a strongly sofic monoid and $A$ is a finite alphabet set, then every injective cellular automaton $\tau \colon A^M \to A^M$ is surjective. As an application, we prove that the monoid algebra of a strongly sofic monoid with coefficients in an arbitrary field is always stably finite. Our results are extensions to strongly sofic monoids of two previously known properties of sofic groups. The first one is the celebrated Gromov-Weiss theorem asserting that every sofic group is surjunctive. The second is the Elek-Szab\'o theorem which says that group algebras of sofic groups satisfy Kaplansky's stable finiteness conjecture.