Rigid actions have zero entropy

TL;DR

This paper demonstrates that rigid actions of free groups have zero or nonpositive entropy, providing new examples of actions with specific entropy properties and invariance under orbit-equivalence.

Contribution

It introduces the first example of an essentially free, ergodic free group action with nonpositive sofic entropy that remains invariant under stable orbit-equivalence.

Findings

Rigid actions have zero Rokhlin entropy.

Rigid actions have nonpositive sofic entropy.

Stable orbit-equivalence preserves nonpositive sofic entropy.

Abstract

Rigid actions have zero Rokhlin entropy and nonpositive sofic entropy. Because rigidity is a stable orbit-equivalence invariant, this provides the first example of an essentially free, ergodic, probability-measure-preserving action of the free group that has nonpositive sofic entropy and any essentially free action stably-orbit-equivalent to it also has nonpositive sofic entropy.