Geodesic
On algebraic properties of topological full groups
Sbornik. Mathematics, Tome 205 (2014) no. 6, pp. 843-861

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We discuss the algebraic structure of the topological full group $[[T]]$ of a Cantor minimal system $(X,T)$. We show that $[[T]]$ has a structure similar to a union of permutational wreath products of the group $\mathbb Z$. This allows us to prove that the topological full groups are locally embeddable into finite groups, give an elementary proof of the fact that the group $[[T]]'$ is infinitely presented, and provide explicit examples of maximal locally finite subgroups of $[[T]]$. We also show that the commutator subgroup $[[T]]'$, which is simple and finitely-generated for minimal subshifts, is decomposable into a product of two locally finite groups, and that $[[T]]$ and $[[T]]'$ possess continuous ergodic invariant random subgroups. Bibliography: 36 titles.
Keywords: full group, Cantor system, finitely generated group
Mots-clés : simple group, amenable group. 
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     author = {R. Grigorchuk and K. Medynets},
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R. Grigorchuk; K. Medynets. On algebraic properties of topological full groups. Sbornik. Mathematics, Tome 205 (2014) no. 6, pp. 843-861. http://geodesic.mathdoc.fr/item/SM_2014_205_6_a3/