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Branch actions and the structure lattice

Fariña-Asategui, Jorge LU and Grigorchuk, Rostislav (2024) In Algebra and Discrete Mathematics 38(2). p.215-232
Abstract

J. S. Wilson proved in 1971 an isomorphism between the structural lattice associated to a group belonging to his second class of groups with every proper quotient finite and the Boolean algebra of clopen subsets of Cantor’s ternary set. In this paper we generalize this isomorphism to the class of branch groups. Moreover, we show that for every faithful branch action of a group G on a spherically homogeneous rooted tree T there is a canonical G-equivariant isomorphism between the Boolean algebra associated to the structure lattice of G and the Boolean algebra of clopen subsets of the boundary of T.

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author
and
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Boolean algebras, branch actions, Stone spaces, the structure lattice
in
Algebra and Discrete Mathematics
volume
38
issue
2
pages
18 pages
publisher
Institute of Applied Mathematics And Mechanics of the National Academy of Sciences of Ukraine
external identifiers
  • scopus:85216806462
ISSN
1726-3255
DOI
10.12958/adm2351
language
English
LU publication?
yes
additional info
Publisher Copyright: © Algebra and Discrete Mathematics Volume 38 (2024). Number 2, pp. 215–232.
id
3bdd6476-9b6a-4c30-a69f-47c8677d73aa
date added to LUP
2025-05-08 15:31:44
date last changed
2025-05-08 17:28:27
@article{3bdd6476-9b6a-4c30-a69f-47c8677d73aa,
  abstract     = {{<p>J. S. Wilson proved in 1971 an isomorphism between the structural lattice associated to a group belonging to his second class of groups with every proper quotient finite and the Boolean algebra of clopen subsets of Cantor’s ternary set. In this paper we generalize this isomorphism to the class of branch groups. Moreover, we show that for every faithful branch action of a group G on a spherically homogeneous rooted tree T there is a canonical G-equivariant isomorphism between the Boolean algebra associated to the structure lattice of G and the Boolean algebra of clopen subsets of the boundary of T.</p>}},
  author       = {{Fariña-Asategui, Jorge and Grigorchuk, Rostislav}},
  issn         = {{1726-3255}},
  keywords     = {{Boolean algebras; branch actions; Stone spaces; the structure lattice}},
  language     = {{eng}},
  number       = {{2}},
  pages        = {{215--232}},
  publisher    = {{Institute of Applied Mathematics And Mechanics of the National Academy of Sciences of Ukraine}},
  series       = {{Algebra and Discrete Mathematics}},
  title        = {{Branch actions and the structure lattice}},
  url          = {{http://dx.doi.org/10.12958/adm2351}},
  doi          = {{10.12958/adm2351}},
  volume       = {{38}},
  year         = {{2024}},
}