On algebraic properties of topological full groups

R. Grigorchuk   ; K. Medynets

Sbornik. Mathematics, Tome 205 (2014) no. 6, pp. 843-861

Résumé

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.

Détail

Keywords: full group, Cantor system, finitely generated group
Mots-clés : simple group, amenable group.

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