Narrow Results

We introduce a class of automorphisms of compact quantum groups (CQGs) which may be thought of as inner automorphisms and explore the behavior of normal subgroups of CQGs under these automorphisms. We also define the notion of center of a CQG and compute the center for several examples. We briefly touch upon the commutator subgroup of a CQG and discuss how its relation with the center can be different from the classical case.

For fixed m, n ≥ 2, we examine the structure of the nth lower central subgroup γ_n(F) of the free group F of rank m with respect to a certain finite chain F = F⁽⁰⁾ > F⁽¹⁾ > ⋯ > F^(l-1) > F^(l) = {1} of free groups in which F^(k) is of finite rank m(k) and is contained in the kth derived subgroup δ_k(F) of F. The derived subgroups δ_k(F/γ_n(F)) of the free nilpotent group F/γ_n(F) are isomorphic to the quotients F^(k)/(F^(k) ∩ γ_n(F)) and admit presentations of the form 〈x_k,1,…,x_k,m(k): γ_(n)(F^(k))〉, where γ_(n)(F^(k)), contained in γ_n(F), is a certain partial lower central subgroup of F^(k). We give a complete description of γ_n(F) as a staggered product Π_{1 ≤ k ≤ l-1}(γ_〈n〉(F^(k))*γ_[n](F^(k)))^F(k+1), where γ_〈n〉(F^(k)) is a free factor of the derived subgroup [F^(k),F^(k)] of F^(k) having countable infinite rank and generated by a certain set of reduced commutators of weight at least n, and γ_[n](F^(k)) is the subgroup generated by a certain finite set of products of non-reduced ordered commutators of weight at least n. There are some far-reaching consequences.

Let ${\mathrm{\text{VB}}}_n$, respectively ${\mathrm{\text{WB}}}_n$ denote the virtual, respectively welded, braid group on $n$-strands. We study their commutator subgroups ${\mathrm{\text{VB}}}_n^{\prime }=[{\mathrm{\text{VB}}}_n,{\mathrm{\text{VB}}}_n]$ and, ${\mathrm{\text{WB}}}_n^{\prime }=[{\mathrm{\text{WB}}}_n,{\mathrm{\text{WB}}}_n]$, respectively. We obtain a set of generators and defining relations for these commutator subgroups. In particular, we prove that ${\mathrm{\text{VB}}}_n^{\prime }$ is finitely generated if and only if $n\ge 4$, and ${\mathrm{\text{WB}}}_n^{\prime }$ is finitely generated for $n\ge 3$. Also, we prove that ${\mathrm{\text{VB}}}_3^{\prime }/{\mathrm{\text{VB}}}_3^{″}={\mathbb{Z}}_3\oplus {\mathbb{Z}}_3\oplus {\mathbb{Z}}_3\oplus {\mathbb{Z}}^{\infty },{\mathrm{\text{VB}}}_4^{\prime }/{\mathrm{\text{VB}}}_4^{″}={\mathbb{Z}}_3\oplus {\mathbb{Z}}_3\oplus {\mathbb{Z}}_3,{\mathrm{\text{WB}}}_3^{\prime }/{\mathrm{\text{WB}}}_3^{″}={\mathbb{Z}}_3\oplus {\mathbb{Z}}_3\oplus {\mathbb{Z}}_3\oplus \mathbb{Z},{\mathrm{\text{WB}}}_4^{\prime }/{\mathrm{\text{WB}}}_4^{″}={\mathbb{Z}}_3,$ and for $n\ge 5$ the commutator subgroups ${\mathrm{\text{VB}}}_n^{\prime }$ and ${\mathrm{\text{WB}}}_n^{\prime }$ are perfect, i.e. the commutator subgroup is equal to the second commutator subgroup.

Giving a presentation of the group of a 2-braid virtual knot or link, we consider the groups of three families of 2-braid virtual knots. Each of them has a certain feature; for example, we can show: for any positive integer N, there exists a virtual knot group with an element of order N. It is known that the collection of the virtual knot groups is the same as that of the ribbon T²-knot groups. Using our examples we discuss the relationship among the virtual knot groups and other knot groups such as ribbon S²-knot groups, S²-knot groups, T²-knot groups, and S³-knot groups.

We show that the commutator subgroup G′ of a classical knot group G need not have subgroups of every finite index, but it will if G′ has a surjective homomorphism to the integers and we give an exact criterion for that to happen. We also give an example of a knotted Sⁿ in Sⁿ⁺² for all n ≥ 2 whose infinite cyclic cover is not simply connected but has no proper finite covers.

Following [6] we consider a knot group G, its commutator subgroup K = [G, G], a finite group Σ and the space Hom(K, Σ) of all representations ρ : K → Σ, endowed with the weak topology. We choose a meridian x ∈ G of the knot and consider the homeomorphism σ_x of Hom(K, Σ) onto itself: σ_xρ(a) = ρ(xax^-1) ∀ a ∈ K, ρ ∈ Hom(K, Σ). As proven in [5], the dynamical system (Hom(K, Σ), σ_x) is a shift of finite type. In the case when Σ is abelian, Hom(K, Σ) is finite.

In this paper we calculate the periods of orbits of (Hom(K, ℤ/p), σ_x), where p is prime, in terms of the roots of the Alexander polynomial of the knot. In the case of two-bridge knots we give a complete description of the set of periods.

The singular braids with $n$ strands, $n\ge 3$, were introduced independently by Baez and Birman. It is known that the monoid formed by the singular braids is embedded in a group that is known as singular braid group, denoted by $S{G}_n$. There has been another generalization of braid groups, denoted by $GV{B}_n$, $n\ge 3$, which was introduced by Fang as a group of symmetries behind quantum quasi-shuffle structures. The group $GV{B}_n$ simultaneously generalizes the classical braid group, as well as the virtual braid group on $n$ strands.

We investigate the commutator subgroups $S{G}_n^{\prime }$ and $GV{B}_n^{\prime }$ of these generalized braid groups. We prove that $S{G}_n^{\prime }$ is finitely generated if and only if $n\ge 5$, and $GV{B}_n^{\prime }$ is finitely generated if and only if $n\ge 4$. Further, we show that both $S{G}_n^{\prime }$ and $GV{B}_n^{\prime }$ are perfect if and only if $n\ge 5$.

For a knot $K$ in $S^3$, let $G(K)$ be the knot group of $K$, $a(K)$ the Ma–Qiu (MQ) index, which is the minimal number of normal generators of the commutator subgroup of $G(K)$, and $m(K)$ the Nakanishi index of $K$, which is the minimal number of generators of the Alexander module of $K$. We generalize the notions for a pair of a group $G$ and its normal subgroup $N$, and we denote them by $a(G,N)$ and $m(G,N),$ respectively. Then it is easy to see $m(G,N)\le a(G,N)$ in general. We also introduce a notion “$\omega$-solvability” for a group that the intersection of all higher commutator subgroups is trivial. Our main theorem is that if $N$ is $\omega$-solvable, then we have $m(G,N)=a(G,N)$. As corollaries, for a fibered knot $K$, we have $m(K)=a(K)$, and we could determine the MQ indices of prime knots up to nine crossings completely.

In this paper, we present a characterization of groups $G$ of order $p^7$, $p$ prime, in which not all elements of the commutator subgroup ${\gamma }_2(G)$ of $G$ are commutators in $G$. In the way, we obtain several structural results on groups of order $p^7$.

The structure of groups for which certain sets of commutator subgroups are finite is investigated. In particular, the relation between such groups and groups with finite conjugacy classes of elements is discussed.

We show the uniform simplicity of the identity component Diff^r(Mⁿ)₀ of the group of C^r diffeomorphisms Diff^r(Mⁿ) (1 ≤ r ≤ ∞, r ≠ n + 1) of the compact connected n-dimensional manifold Mⁿ with handle decomposition without handles of the middle index n/2. More precisely, for any elements f and g of such Diff^r(Mⁿ)₀ \ {id}, f can be written as a product of at most 16n + 28 conjugates of g or g^-1, which we denote by f ∈ (C_g)¹⁶ⁿ⁺²⁸. We have better estimates for several manifolds. For the n-dimensional sphere Sⁿ, for any elements f and g of Diff^r(Sⁿ)₀ \ {id} (1 ≤ r ≤ ∞, r ≠ n + 1), f ∈ (C_g)¹², and for a compact connected 3-manifold M³, for any elements f and g of Diff^r(M³)₀ \ {id} (1 ≤ r ≤ ∞, r ≠ 4), f ∈ (C_g)⁴⁴.