A topological proof in group theory

1963 ◽  
Vol 59 (2) ◽  
pp. 277-282
Author(s):  
D. E. Cohen
Keyword(s):  
Group Theory ◽  
Free Group ◽  
Related Result ◽  
Normal Closure ◽  
Single Element ◽  
Covering Spaces ◽  
Topological Theory ◽  
Topological Proof ◽  
Identity Theorem

The topological theory of covering spaces may be used to prove results in group theory, for instance, the Kuros-Reidemeister-Schreier theorem (1). It seems likely that such methods can be applied to prove the Freiheitsatz (4) and the identity theorem (3), and also perhaps Lyndon's conjecture, that the normal closure in a free group F of a, single element r is freely generated by conjugates of r. However, although these problems may easily be stated in topological terms, no such proof is at present known. In this paper we prove a related result.

scholarly journals The normal closure of a power of a half-twist has infinite index in the mapping class group of a punctured sphere

2019 ◽  
Vol 11 (02) ◽  
pp. 273-292
Author(s):  
Charalampos Stylianakis
Keyword(s):  
Free Group ◽  
Braid Group ◽  
Mapping Class Group ◽  
Abelian Subgroup ◽  
Infinite Order ◽  
Class Group ◽  
Normal Closure ◽  
Mapping Class ◽  
Infinite Index ◽  
Half Twist

In this paper we show that the normal closure of the [Formula: see text]th power of a half-twist has infinite index in the mapping class group of a punctured sphere if [Formula: see text] is at least five. Furthermore, in some cases we prove that the quotient of the mapping class group of the punctured sphere by the normal closure of a power of a half-twist contains a free abelian subgroup. As a corollary we prove that the quotient of the hyperelliptic mapping class group of a surface of genus at least two by the normal closure of the [Formula: see text]th power of a Dehn twist has infinite order, and for some integers [Formula: see text] the quotient contains a free group. As a second corollary we recover a result of Coxeter: the normal closure of the [Formula: see text]th power of a half-twist in the braid group of at least four strands has infinite index. Our method is to reformulate the Jones representation of the mapping class group of a punctured sphere, using the action of Hecke algebras on [Formula: see text]-graphs, as introduced by Kazhdan–Lusztig.

One-relator surface groups

1990 ◽  
Vol 108 (3) ◽  
pp. 467-474 ◽  
Author(s):  
John Hempel
Keyword(s):  
Normal Subgroup ◽  
Single Point ◽  
Surface Group ◽  
Compact Surface ◽  
Heegaard Splitting ◽  
Normal Closure ◽  
Single Element ◽  
Homotopy Classes ◽  
Simple Loop ◽  
Proper Power

For X a subset of a group G, the smallest normal subgroup of G which contains X is called the normal closure of X and is denoted by ngp (X; G) or simply by ngp (X) if there is no possibility of ambiguity. By a surface group we mean the fundamental group of a compact surface. We are interested in determining when a normal subgroup of a surface group contains a simple loop – the homotopy class of an embedding of S1 in the surface, or more generally, a power of a simple loop. This is significant to the study of 3-manifolds since a Heegaard splitting of a 3-manifold is reducible (cf. [2]) if and only if the kernel of the corresponding splitting homomorphism contains a simple loop. We give an answer in the case that the normal subgroup is the normal closure ngp (α) of a single element α: if ngp (α) contains a (power of a) simple loop β then α is homotopic to a (power of a) simple loop and β±1 is homotopic either to (a power of) α or to the commutator [α, γ] of a with some simple loop γ meeting a transversely in a single point. This implies that if a is not homotopic to a power of a simple loop, then the quotient map π1(S) → π1(S)/ngp (α) does not factor through a group with more than one end. In the process we show that π1(S)/ngp (α) is locally indicable if and only if α is not a proper power and that α always lifts to a simple loop in the covering space Sα of S corresponding to ngp (α). We also obtain some estimates on the minimal number of double points in certain homotopy classes of loops.

A Note on Fibonacci Type Groups

1975 ◽  
Vol 18 (2) ◽  
pp. 173-175 ◽  
Author(s):  
C. M. Campbell ◽  
E. F. Robertson
Keyword(s):  
Free Group ◽  
Normal Closure ◽  
Index Set ◽  
Congruence Classes

Let Fn be the free group on {ai:i ∊ ℤ n} where the set of congruence classes mod n is used as an index set for the generators. The permutation (1, 2, 3, …, n) of ℤn induces an automorphism θ of Fn by permuting the subscripts of the generators. Suppose w is a word in Fn and let N(w) denote the normal closure of {wθi-1:l ≤i≤n}. Define the group Gn(w) by Gn(w)=Fn/N(w) and call wdi-1=l the relation (i) of Gn(w).

The Schreier Technique for Subalgebras of a Free Lie Algebra

1997 ◽  
Vol 49 (3) ◽  
pp. 600-616 ◽  
Author(s):  
Shmuel Rosset ◽  
Alon Wasserman
Keyword(s):  
Group Theory ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Free Group ◽  
Finite Codimension ◽  
Free Lie Algebra ◽  
Finitely Generated ◽  
Free Lie Algebras

AbstractIn group theory Schreier's technique provides a basis for a subgroup of a free group. In this paper an analogue is developed for free Lie algebras. It hinges on the idea of cutting a Hall set into two parts. Using it, we show that proper subalgebras of finite codimension are not finitely generated and, following M. Hall, that a finitely generated subalgebra is a free factor of a subalgebra of finite codimension.

FREE GROUP ROOTS OF akbl AND [ ak,b]

2000 ◽  
Vol 10 (03) ◽  
pp. 339-347
Author(s):  
JAMES McCOOL
Keyword(s):  
Finite Number ◽  
Free Group ◽  
Conjugacy Classes ◽  
Normal Closure

Let F be a free group with basis a, b, …, and let u, w ∈ F . We say that u is a root of w if w belongs to the normal closure of u in F . We show that if w is of the form akbl then w has only a finite number of conjugacy classes of roots, and that given k and l the set of roots of w can be algorithmically determined. We also classify the roots of elements of the form akba-kb-1.

scholarly journals Almost congruence extension property for subgroups of free groups

2018 ◽  
Vol 21 (1) ◽  
pp. 125-146
Author(s):  
Lev Glebsky ◽  
Nevarez Nieto Saul
Keyword(s):  
Normal Subgroup ◽  
Free Group ◽  
Free Groups ◽  
Extension Property ◽  
Normal Closure ◽  
Sufficient Condition ◽  
Congruence Extension Property ◽  
Finitely Generated ◽  
Finite Set

AbstractLetHbe a subgroup ofFand{\langle\kern-1.422638pt\langle H\rangle\kern-1.422638pt\rangle_{F}}the normal closure ofHinF. We say thatHhas the Almost Congruence Extension Property (ACEP) inFif there is a finite set of nontrivial elements{\digamma\subset H}such that for any normal subgroupNofHone has{H\cap\langle\kern-1.422638pt\langle N\rangle\kern-1.422638pt\rangle_{F}=N}whenever{N\cap\digamma=\emptyset}. In this paper, we provide a sufficient condition for a subgroup of a free group to not possess ACEP. It also shows that any finitely generated subgroup of a free group satisfies some generalization of ACEP.

Systems of equations and generalized characters in groups

1970 ◽  
Vol 22 (5) ◽  
pp. 1040-1046 ◽  
Author(s):  
I. M. Isaacs
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Free Group ◽  
Equivalence Relation ◽  
Elementary Proof ◽  
Systems Of Equations ◽  
Recent Application ◽  
Arbitrary Group ◽  
Frobenius Theorem ◽  
A Finite Group

Let F be the free group on n generators x1, …, Xn and let G be an arbitrary group. An element ω ∈ F determines a function x → ω(x) from n-tuples x = (x1, x2, …, xn) ∈ Gn into G. In a recent paper [5] Solomon showed that if ω1, ω2, …, ωm ∈ F with m < n, and K1, …, Km are conjugacy classes of a finite group G, then the number of x ∈ Gn with ωi(x) ∈ Ki for each i, is divisible by |G|. Solomon proved this by constructing a suitable equivalence relation on Gn.Another recent application of an unusual equivalence relation in group theory is in Brauer's paper [1], where he gives an elementary proof of the Frobenius theorem on solutions of xk = 1 in a group.

The Ping-Pong Lemma

2017 ◽  
Author(s):  
Johanna Mangahas
Keyword(s):  
Group Theory ◽  
Free Group ◽  
Group Action ◽  
Metric Space ◽  
Hyperbolic Geometry ◽  
Free Groups ◽  
Central Place ◽  
Geometric Group Theory ◽  
Research Projects ◽  
Ping Pong

This chapter considers an identifying feature of free groups: their ability to play ping-pong. In mathematics, you may encounter a group without immediately knowing which group it is. Fortunately, you can tell a group by how it acts. That is, a good group action (for example, action by isometries on a metric space) can reveal a lot about the group itself. This theme occupies a central place in geometric group theory. The ping-pong lemma, also dubbed Schottky lemma or Klein's criterion, gives a set of circumstances for identifying whether a group is a free group. The chapter first presents the statement, proof, and first examples using ping-pong before discussing ping-pong with Möbius transformations and hyperbolic geometry. Exercises and research projects are included.

scholarly journals Non-triviality of some one-relator products of three groups

2016 ◽  
Vol 26 (03) ◽  
pp. 533-550
Author(s):  
Ihechukwu Chinyere ◽  
James Howie
Keyword(s):  
Free Product ◽  
Normal Closure ◽  
Single Element

In this paper we study a group [Formula: see text] which is the quotient of a free product of three nontrivial groups by the normal closure of a single element. In particular, we show that if the relator has length at most eight, then [Formula: see text] is nontrivial. In the case where the factors are cyclic, we prove the stronger result that at least one of the factors embeds in [Formula: see text].

scholarly journals Low dimensional homotopy for monoids II: groups

1999 ◽  
Vol 41 (1) ◽  
pp. 1-11 ◽  
Author(s):  
STEPHEN J. PRIDE
Keyword(s):  
Free Group ◽  
Equivalence Classes ◽  
Normal Closure ◽  
Group Presentation ◽  
One To One ◽  
Low Dimensional ◽  
Reduced Words

Consider a group presentation: $$\hat{[Pscr ]}\tfrm{=<\tfbf{x};}\tfbf{r}\tfrm{>}$$. Here x is a set and r is a set of non-empty, cyclically reduced words on the alphabet x ∪ x−1 (where x−1 is a set in one-to-one correspondence x[harr ]x−1 with x). We assume throughout that $\hat{[Pscr ]}$ is finite. Let $\hat{F}$ be the free group on x (thus $\hat{F}$ consists of free equivalence classes [W] of word on x∪x−1), and let N be the normal closure of {[R] : R∈r} in $\hat{F}$. Then the group G=G($\hat{[Pscr ]}$) defined by $\hat{[Pscr ]}$ is $\hat{F}\tfrm{/}N$. We will write W1 =GW2 if [W1]N=[W2]N.

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