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JOURNALS // Matematicheskii Sbornik // Archive

Mat. Sb., 2014 Volume 205, Number 6, Pages 87–108 (Mi sm8257)

This article is cited in 20 papers

On algebraic properties of topological full groups

R. Grigorchukab, K. Medynetsc

a Steklov Mathematical Institute of Russian Academy of Sciences
b Texas A&M University
c United States Naval Academy

Abstract: 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, simple group, amenable group.

UDC: 512.543+512.544+517.987

MSC: 20F65, 37B05

Received: 11.06.2013 and 10.02.2014

DOI: 10.4213/sm8257


 English version:
Sbornik: Mathematics, 2014, 205:6, 843–861

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© Steklov Math. Inst. of RAS, 2024