On Algebraic Properties of Topological Full Groups

2011/05/04 by Grigorchuk, Rostislav, Medynets, Konstantin
#20E32 #43A07 #54H15 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1105.0719

Abstract

In the paper we discuss the algebraic structure of topological full group [[T]] of a Cantor minimal system (X,T). We show that the topological full group [[T]] has the structure similar to a union of permutational wreath products of group \mathbb Z. This allows us to prove that the topological full groups are locally embeddable into finite groups; give an elmentary proof of the fact that 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 the groups [[T]] and [[T]]' possess continuous ergodic invariant random subgroups.

Related

• Some remarks on topological full groups of Cantor minimal systems
• Some remarks on topological full groups of Cantor minimal systems II
• Full groups, flip conjugacy, and orbit equivalence of Cantor minimal systems
• Topological full groups of minimal subshifts with subgroups of intermediate growth
• Topological Full Groups