scholarly journals PROFINITE COMPLETION OF GRIGORCHUK'S GROUP IS NOT FINITELY PRESENTED

2012 ◽  
Vol 22 (05) ◽  
pp. 1250045
Author(s):  
MUSTAFA GÖKHAN BENLI
Keyword(s):  
Profinite Group ◽  
Profinite Completion ◽  
Infinite Dimensional ◽  
Grigorchuk Group ◽  
Schur Multipliers ◽  
Finite Quotients ◽  
Minimal Presentations ◽  
Finitely Presented

In this paper we prove that the profinite completion [Formula: see text] of the Grigorchuk group [Formula: see text] is not finitely presented as a profinite group. We obtain this result by showing that [Formula: see text] is infinite dimensional. Also several results are proven about the finite quotients [Formula: see text] including minimal presentations and Schur Multipliers.

scholarly journals Metabelian groups with the same finite quotients

1974 ◽  
Vol 11 (1) ◽  
pp. 115-120 ◽  
Author(s):  
P.F. Pickel
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Metabelian Group ◽  
Metabelian Groups ◽  
Isomorphism Classes ◽  
Split Extensions ◽  
Finite Quotients ◽  
Finitely Presented

Let F(G) denote the set of isomorphism classes of finite quotients of the group G. Two groups G and H are said to have the same finite quotients if F(G) = F(H). We construct infinitely many nonisomorphic finitely presented metabelian groups with the same finite quotients, using modules over a suitably chosen ring. These groups also give an example of infinitely many nonisomorphic split extensions of a fixed finitely presented metabelian group by a fixed finite abelian group, all having the same finite quotients.

Minimal Presentations and Embedding into Inefficient Semigroups

2005 ◽  
Vol 12 (01) ◽  
pp. 59-65 ◽  
Author(s):  
H. Ayık ◽  
M. Minisker ◽  
B. Vatansever
Keyword(s):  
Direct Product ◽  
Minimal Presentations ◽  
Finitely Presented ◽  
Free Semilattice

In this paper, we show that CLn, the chain with n elements, is efficient and that the direct product CLm × CLn is inefficient. Moreover, we embed any finitely presented semigroup S into an inefficient semigroup, namely, the semigroup S ⋃ SLn, where SLn is the free semilattice of rank n.

scholarly journals The weight of nonstrongly complete profinite groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Tamar Bar-On
Keyword(s):  
Profinite Group ◽  
Profinite Groups ◽  
Profinite Completion ◽  
Local Weight

Abstract We compute the local weight of the completion of a nonstrongly complete profinite group and conclude that, if a profinite group is abstractly isomorphic to its own profinite completion, then they are equal. The local weights of all the groups in the tower of completions are computed as well.

scholarly journals On strongly just infinite profinite branch groups

2017 ◽  
Vol 20 (1) ◽  
pp. 1-32 ◽  
Author(s):  
François Le Maître ◽  
Phillip Wesolek
Keyword(s):  
Normal Subgroup ◽  
Simple Groups ◽  
Continuity Property ◽  
Compact Groups ◽  
Automatic Continuity ◽  
Profinite Completion ◽  
Locally Compact ◽  
Grigorchuk Group ◽  
Continuity Properties ◽  
Additional Assumptions

AbstractFor profinite branch groups, we first demonstrate the equivalence of the Bergman property, uncountable cofinality, Cayley boundedness, the countable index property, and the condition that every non-trivial normal subgroup is open; compact groups enjoying the last condition are called strongly just infinite. For strongly just infinite profinite branch groups with mild additional assumptions, we verify the invariant automatic continuity property and the locally compact automatic continuity property. Examples are then presented, including the profinite completion of the first Grigorchuk group. As an application, we show that many Burger–Mozes universal simple groups enjoy several automatic continuity properties.

scholarly journals Hyperbolic and cubical rigidities of Thompson’s group V

2019 ◽  
Vol 22 (2) ◽  
pp. 313-345 ◽  
Author(s):  
Anthony Genevois
Keyword(s):  
General Criterion ◽  
Isometric Action ◽  
Group V ◽  
Cube Complex ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Fixed Point Properties ◽  
Thompson’S Group ◽  
Cube Complexes ◽  
Finitely Presented

Abstract In this article, we state and prove a general criterion allowing us to show that some groups are hyperbolically elementary, meaning that every isometric action of one of these groups on a Gromov-hyperbolic space either fixes a point at infinity, or stabilises a pair of points at infinity, or has bounded orbits. Also, we show how such a hyperbolic rigidity leads to fixed-point properties on finite-dimensional CAT(0) cube complexes. As an application, we prove that Thompson’s group V is hyperbolically elementary, and we deduce that it satisfies Property {({\rm FW}_{\infty})} , i.e., every isometric action of V on a finite-dimensional CAT(0) cube complex fixes a point. It provides the first example of a (finitely presented) group acting properly on an infinite-dimensional CAT(0) cube complex such that all its actions on finite-dimensional CAT(0) cube complexes have global fixed points.

A Remark on Finitely Presented Infinite Dimensional Algebras

1990 ◽  
Vol 108 (3) ◽  
pp. 633 ◽  
Author(s):  
Gilbert Baumslag ◽  
Peter B. Shalen
Keyword(s):  
Infinite Dimensional ◽  
Finitely Presented
Sign in / Sign up

Export Citation Format

Share Document