Boundary actions of outer automorphism groups of Thompson-like groups

For every Cuntz–Krieger groupoid, we show that there is a topologically free boundary action of the outer automorphism group of its topological full group on the Hilbert cube. In particular, these outer automorphism groups, including the outer automorphism groups of all Higman–Thompson groups, are C*-simple.

💡 Research Summary

The paper investigates the outer automorphism groups of topological full groups associated with Cuntz–Krieger groupoids, a class of étale groupoids that model Cuntz–Krieger C*-algebras arising from irreducible shifts of finite type. The authors prove that for every such groupoid (G_E), the outer automorphism group (\operatorname{Out}(F(G_E))) admits a topologically free boundary action on the Hilbert cube. This action is realized on the projectivised space of positive, shift‑invariant Radon measures on the underlying shift space, which is homeomorphic to the Hilbert cube. By the Kalantar–Kennedy characterization of C*-simplicity, the existence of a topologically free boundary action implies that (\operatorname{Out}(F(G_E))) is C*-simple. Consequently, the outer automorphism groups of all Higman–Thompson groups (V_{n,r}) are C*-simple.

The proof proceeds in three main stages. First, the authors identify (\operatorname{Out}(G_E)) with (\operatorname{Out}(F(G_E))) using Matui’s work on topological full groups. They then describe automorphisms of the groupoid via continuous orbit equivalences (COEs) of the underlying one‑sided shift of finite type, relying heavily on the theory developed by Matsumoto and Matsumoto–Matui. The action on the measure space is induced from the natural action on the first cohomology group (H^1(G_E)), which coincides with the dynamical cohomology of the shift.

The second stage establishes two dynamical properties of the induced action on the set of primitive cycles of the defining graph (E). Using the Boyle–Krieger classification of shift automorphisms and Matsumoto’s strong COEs, the authors prove faithfulness: distinct elements of (\operatorname{Out}(G_E)) act differently on the set of primitive cycles. They then prove transitivity by a combinatorial “word problem” argument, showing that any two primitive cycles can be related by an outer automorphism constructed via marker COEs.

The third and most technical stage proves strong proximality of the action on the projectivised measure space. The authors introduce marker COEs, inspired by marker automorphisms in symbolic dynamics, and verify intricate overlap conditions to ensure these COEs are well‑defined. Lemma 6.9 (the “key lemma”) shows that sequences of homeomorphisms that are “very good” in a partial equicontinuity sense can be composed to force convergence of images of points to prescribed limits. By constructing appropriate sequences of marker COEs and using a refined classification theorem for Cuntz–Krieger groupoids (ensuring every groupoid is isomorphic to one coming from a primitive graph), they obtain the required convergence of arbitrary probability measures to point masses, establishing strong proximality.

Combining faithfulness, transitivity, and strong proximality yields a topologically free boundary action, and thus C*-simplicity of (\operatorname{Out}(F(G_E))). As a corollary, the authors compute the amenable radical of the outer automorphism groups of the simple commutator subgroups (D(G_E)). They show it is isomorphic to a direct sum involving the cokernel and kernel of (I-A_E^t) (where (A_E) is the adjacency matrix), and they give a precise criterion: (\operatorname{Out}(