HIGMAN-THOMPSON GROUPS FROM SELF-SIMILAR GROUPOID ACTIONS

The paper proves that for an effective groupoid of germs Gp(G,E) arising from a self-similar groupoid action, the Higman–Thompson-type group VE(G) is isomorphic to the topological full group vGp(G,E)^ω. It does so by sending a G-table to a full bisection U = ⊔ Z(α_i,g_i,β_i) and then to the induced homeomorphism π_U, noting injectivity and using effectivity to assert uniqueness of the full bisection representing a given π ∈ vG for surjectivity . The model’s solution builds exactly the same correspondence: it constructs the basic compact-open bisections U(α,g,β)=Z(α,g,β), proves that splitting a column corresponds to a disjoint union of basic bisections, checks multiplicativity by refining partitions, and shows any full bisection decomposes into a finite disjoint union of basics, yielding a table; this uses the groupoid’s topology and the Z(α,g,β;U) basis property together with the table splitting rule and the definition of the topological full group . The only difference is emphasis: the paper phrases effectivity as providing uniqueness of the representing full bisection, while the model highlights effectivity for canonicity of the inverse construction. Substantively, the proofs coincide.