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Mat. Sb., 2014, Volume 205, Number 6, Pages 87–108 (Mi msb8257)  

This article is cited in 10 scientific papers (total in 10 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.

Funding Agency Grant Number
National Science Foundation DMS-1207699
Office of Naval Research N001613WX20992

Author to whom correspondence should be addressed


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

Bibliographic databases:

UDC: 512.543+512.544+517.987
MSC: 20F65, 37B05
Received: 11.06.2013 and 10.02.2014

Citation: R. Grigorchuk, K. Medynets, “On algebraic properties of topological full groups”, Mat. Sb., 205:6 (2014), 87–108; Sb. Math., 205:6 (2014), 843–861

Citation in format AMSBIB
\by R.~Grigorchuk, K.~Medynets
\paper On algebraic properties of topological full groups
\jour Mat. Sb.
\yr 2014
\vol 205
\issue 6
\pages 87--108
\jour Sb. Math.
\yr 2014
\vol 205
\issue 6
\pages 843--861

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    This publication is cited in the following articles:
    1. K. Juschenko, M. de la Salle, “Invariant means for the wobbling group”, Bull. Belg. Math. Soc. Simon Stevin, 22:2 (2015), 281–290  crossref  mathscinet  zmath  isi
    2. M. I. Cortez, K. Medynets, “Orbit equivalence rigidity of equicontinuous systems”, J. Lond. Math. Soc. (2), 94:2 (2016), 545–556  crossref  mathscinet  zmath  isi  scopus
    3. H. Matui, “Topological full groups of Étale groupoids”, Operator Algebras and Applications, Abel Symposia, 12, eds. T. Carlsen, N. Larsen, S. Neshveyev, C. Skau, Springer, Cham, 2016, 197–224  crossref  mathscinet  zmath  scopus
    4. T. Ibarlucía, J. Melleray, “Full groups of minimal homeomorphisms and Baire category methods”, Ergod. Theory Dyn. Syst., 36:2 (2016), 550–573  crossref  mathscinet  zmath  isi  scopus
    5. Ya. Glasner, “Invariant random subgroups of linear groups”, Israel J. Math., 219:1 (2017), 215–270  crossref  mathscinet  zmath  isi  scopus
    6. R. Grigorchuk, K. Medynets, “Presentations of topological full groups by generators and relations”, J. Algebra, 500:SI (2018), 46–68  crossref  mathscinet  zmath  isi  scopus
    7. M. Boyle, S. Chuysurichay, “The mapping class group of a shift of finite type”, J. Mod. Dyn., 13 (2018), 115–145  crossref  mathscinet  zmath  isi
    8. Alekseev V., Finn-Sell M., “Sofic Boundaries of Groups and Coarse Geometry of Sofic Approximations”, Group. Geom. Dyn., 13:1 (2019), 191–234  crossref  mathscinet  zmath  isi
    9. Thomas S., “Topological Full Groups of Minimal Subshifts and the Classification Problem For Finitely Generated Complete Groups”, Group. Geom. Dyn., 13:1 (2019), 327–347  crossref  mathscinet  zmath  isi
    10. Scarparo E., “On the C-Algebra Generated By the Koopman Representation of a Topological Full Group”, Bull. Belg. Math. Soc.-Simon Steven, 26:3 (2019), 469–479  crossref  mathscinet  zmath  isi
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