Simple groups of dynamical origin

Abstract

Abstract. We associate with everyétale groupoid G two normal subgroups S(G) and A(G) of the topological full group of G, which are analogs of the symmetric and alternating groups. We prove that if G is a minimal groupoid of germs (e.g., of a group action), then A(G) is simple and is contained in every non-trivial normal subgroup of the full group. We show that if G is expansive (e.g., is the groupoid of germs of an expansive action of a group), then A(G) is finitely generated. We also show that S(G)/A(G) is a quotient of H 0 (G, Z/2Z).

On a measurable analogue of small topological full groups

“…Apart from Seward's characterization of subshifts, another key ingredient in the proof is Nekrashevych's work on alternating topological full groups [Nek17], which provides dense finitely generated subgroups in the derived L 1 full groups. The reader is refered to Section 2.9 for the definition of alternating topological full group and Section 3 for the proof of their density in L 1 full groups.…”

Topological Full Groups of Étale Groupoids

“…The proposition above can be also thought of as an immediate consequence of the following theorem of M. Rubin [37] (see also [3,Section 9], [33,Section 3.3]). Let X be a topological space.…”

“…Notice that it is not clear at all if this is really well-defined. One can find a proof of the well-definedness in [33]. [26,Section 3]).…”

Tarski monoids: Matui’s spatial realization theorem

“…Matui's spatial realization theorem [19,Theorem 3.10] is phrased entirely in the language ofétale groupoids. By applying non-commutative Stone duality [13], the equivalence of (1) and (2) below is immediate and that of (1) and (3) follows from [20] (who appeals to to [24]). The inverse monoid approach described in our paper is entirely consonant with the approach adopted by Matui.…”