scholarly journals FINITELY GENERATED INFINITE SIMPLE GROUPS OF INFINITE SQUARE WIDTH AND VANISHING STABLE COMMUTATOR LENGTH

2010 ◽  
Vol 02 (03) ◽  
pp. 341-384 ◽  
Author(s):  
ALEXEY MURANOV
Keyword(s):  
Simple Groups ◽  
Finitely Generated ◽  
Invariant Norm ◽  
Stable Commutator Length ◽  
Commutator Length ◽  
Commutator Width

It is shown that there exist finitely generated infinite simple groups of infinite commutator width and infinite square width on which there exists no stably unbounded conjugation-invariant norm, and in particular stable commutator length vanishes. Moreover, a recursive presentation of such a group with decidable word and conjugacy problems is constructed.

scholarly journals FINITELY GENERATED INFINITE SIMPLE GROUPS OF INFINITE COMMUTATOR WIDTH

2007 ◽  
Vol 17 (03) ◽  
pp. 607-659 ◽  
Author(s):  
ALEXEY MURANOV
Keyword(s):  
Simple Group ◽  
Simple Groups ◽  
Finitely Generated ◽  
Commutator Width

It is shown that there exists a finitely generated infinite simple group of infinite commutator width, and that the commutator width of a finitely generated infinite boundedly simple group can be arbitrarily large. Besides, such groups can be constructed with decidable word and conjugacy problems.

On Commutator Length and Square Length of the Wreath Product of a Group by a Finitely Generated Abelian Group

2010 ◽  
Vol 17 (spec01) ◽  
pp. 799-802 ◽  
Author(s):  
Mehri Akhavan-Malayeri
Keyword(s):  
Abelian Group ◽  
Wreath Product ◽  
Finitely Generated ◽  
Commutator Length

Let W = G ≀ H be the wreath product of G by an n-generator abelian group H. We prove that every element of W′ is a product of at most n+2 commutators, and every element of W2 is a product of at most 3n+4 squares in W. This generalizes our previous result.

Simple groups and irreducible lattices in wreath products

2020 ◽  
pp. 1-12 ◽  
Author(s):  
ADRIEN LE BOUDEC
Keyword(s):  
Finite Group ◽  
Wreath Product ◽  
Free Group ◽  
Wreath Products ◽  
Simple Groups ◽  
Finitely Generated ◽  
Locally Compact ◽  
Regular Tree ◽  
Finitely Generated Groups ◽  
A Finite Group

We consider the finitely generated groups acting on a regular tree with almost prescribed local action. We show that these groups embed as cocompact irreducible lattices in some locally compact wreath products. This provides examples of finitely generated simple groups quasi-isometric to a wreath product $C\wr F$ , where $C$ is a finite group and $F$ a non-abelian free group.

scholarly journals Surface groups, infinite generating sets, and stable commutator length

2019 ◽  
Vol 150 (5) ◽  
pp. 2379-2386
Author(s):  
Dan Margalit ◽  
Andrew Putman
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Surface Group ◽  
Mapping Class ◽  
Surface Groups ◽  
Closed Curves ◽  
Generating Set ◽  
Generating Sets ◽  
Stable Commutator Length ◽  
Commutator Length

AbstractWe give a new proof of a theorem of D. Calegari that says that the Cayley graph of a surface group with respect to any generating set lying in finitely many mapping class group orbits has infinite diameter. This applies, for instance, to the generating set consisting of all simple closed curves.

scholarly journals Stable commutator length in amalgamated free products

2015 ◽  
Vol 07 (04) ◽  
pp. 693-717 ◽  
Author(s):  
Tim Susse
Keyword(s):  
Geometric Model ◽  
Free Products ◽  
Cyclic Groups ◽  
Amalgamated Free Products ◽  
Torus Knot ◽  
Knot Complements ◽  
Stable Commutator Length ◽  
Free Abelian Groups ◽  
Commutator Length ◽  
Boundary Mapping

We show that stable commutator length is rational on free products of free abelian groups amalgamated over ℤk, a class of groups containing the fundamental groups of all torus knot complements. We consider a geometric model for these groups and parametrize all surfaces with specified boundary mapping to this space. Using this work we provide a topological algorithm to compute stable commutator length in these groups. Further, we use the methods developed to show that in free products of cyclic groups the stable commutator length of a fixed word varies quasirationally in the orders of the free factors.

T-systems of certain finite simple groups

1993 ◽  
Vol 113 (1) ◽  
pp. 9-22 ◽  
Author(s):  
Martin J. Evans
Keyword(s):  
Free Group ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Number Of Generators

Let Fn be the free group of rank n freely generated by x1, x2,…, xn and write d(G) for the minimal number of generators of the finitely generated group G.

POLYNOMIAL INDEX GROWTH GROUPS

2000 ◽  
Vol 10 (06) ◽  
pp. 773-782 ◽  
Author(s):  
ANTAL BALOG ◽  
LÁSZLÓ PYBER ◽  
AVINOAM MANN
Keyword(s):  
Finite Groups ◽  
Simple Groups ◽  
Finitely Generated ◽  
Constant Power ◽  
Residually Finite ◽  
Almost Simple Groups ◽  
Small Constant ◽  
Residually Finite Groups ◽  
Strengthened Form

We show that the order of a finite simple of Lie type is bounded by a small constant power of its exponent. This confirms, in a strengthened form, a conjecture of Vaughan-Lee and Zel'manov on the order and exponent of almost simple groups. We also obtain various structural restrictions on groups of polynomial index growth. Combining the above results we construct finitely generated residually finite groups of polynomial index growth which are neither linear nor boundedly generated. This answers questions of Segal and Platonov–Rapinchuk respectively. A further question of Platonov–Rapinchuk concerning a weakened polynomial index growth assumption is also answered.

scholarly journals Simple groups of dynamical origin

2017 ◽  
Vol 39 (3) ◽  
pp. 707-732 ◽  
Author(s):  
V. NEKRASHEVYCH
Keyword(s):  
Normal Subgroup ◽  
Group Action ◽  
Simple Groups ◽  
Full Group ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Alternating Groups ◽  
Étale Groupoid ◽  
Dynamical Origin

We associate with every étale groupoid $\mathfrak{G}$ two normal subgroups $\mathsf{S}(\mathfrak{G})$ and $\mathsf{A}(\mathfrak{G})$ of the topological full group of $\mathfrak{G}$, which are analogs of the symmetric and alternating groups. We prove that if $\mathfrak{G}$ is a minimal groupoid of germs (e.g., of a group action), then $\mathsf{A}(\mathfrak{G})$ is simple and is contained in every non-trivial normal subgroup of the full group. We show that if $\mathfrak{G}$ is expansive (e.g., is the groupoid of germs of an expansive action of a group), then $\mathsf{A}(\mathfrak{G})$ is finitely generated. We also show that $\mathsf{S}(\mathfrak{G})/\mathsf{A}(\mathfrak{G})$ is a quotient of $H_{0}(\mathfrak{G},\mathbb{Z}/2\mathbb{Z})$.

scholarly journals Palindromic words in simple groups

2015 ◽  
Vol 25 (03) ◽  
pp. 439-444 ◽  
Author(s):  
Elisabeth Fink ◽  
Andreas Thom
Keyword(s):  
Simple Group ◽  
Infinite Group ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Generating Set ◽  
Commutator Width

A palindrome is a word that reads the same left-to-right as right-to-left. We show that every simple group has a finite generating set X, such that every element of it can be written as a palindrome in the letters of X. Moreover, every simple group has palindromic width pw(G, X) = 1, where X only differs by at most one additional generator from any given generating set. On the contrary, we prove that all non-abelian finite simple groups G also have a generating set S with pw(G, S) > 1. As a by-product of our work we also obtain that every just-infinite group has finite palindromic width with respect to a finite generating set. This provides first examples of groups with finite palindromic width but infinite commutator width.

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