Narrow Results

The balanced superelliptic mapping class group is the normalizer of the transformation group of the balanced superelliptic covering space in the mapping class group of the total surface. We give finite presentations for the balanced superelliptic mapping class groups of closed surfaces, surfaces with one marked point, and surfaces with one boundary component. To give these presentations, we construct finite presentations for corresponding liftable mapping class groups in a different generating set from Ghaswala–Winarski’s presentation in [9].

The balanced superelliptic handlebody group is the normalizer of the transformation group of the balanced superelliptic covering space in the handlebody group of the total space. We give a finite presentation for the balanced superelliptic handlebody group. To give this presentation, we construct a finite presentation for the liftable Hilden group.

Let m,n ∈ ℕ. We define to be the set of (n+m)-braids of the sphere whose associated permutation lies in the subgroup S_n × S_m of the symmetric group S_n+m on n+m letters. In a previous paper [13], we showed that if n ≥ 3, then there exists the following generalisation of the Fadell–Neuwirth short exact sequence:

Our main results are as follows: if n = 1 (respectively, n = 2) then the homomorphism p_* and the fibration p admit (respectively, do not admit) a section. If n = 3, then p_* and p admit a section if and only if m ≡ 0,2 (mod 3). If n ≥ 4, we show that if p_* and p admit a section then m ≡ ε₁(n - 1)(n - 2) - ε₂n(n - 2) (mod n(n - 1)(n - 2)), where ε₁,ε₂ ∈ {0,1}.

Finally, we show that is generated by two of its torsion elements.

Based on the Kauffman bracket at A = e^iπ/4, we define an invariant for a special type of n-punctured ball tangles. The invariant Fⁿ takes values in the set PM_{2 × 2ⁿ}(ℤ) of 2 × 2ⁿ matrices over ℤ modulo the scalar multiplication of ±1. We provide the formula to compute the invariant of the k₁ + ⋯ + k_n-punctured ball tangle composed of given n, k₁, …, k_n-punctured ball tangles. Also, we define the horizontal and the vertical connect sums of punctured ball tangles and provide the formulae for their invariants from those of given punctured ball tangles. In addition, we introduce the elementary operations on the class ST of 1-punctured ball tangles, called spherical tangles. The elementary operations on ST induce the operations on PM_{2 × 2}(ℤ), also called the elementary operations. We show that the group generated by the elementary operations on PM_{2 × 2}(ℤ) is isomorphic to a Coxeter group.

In this paper, we define an invariant of free links valued in a free product of some copies of ${\mathbb{Z}}_2$. In [Non-Reidemeister knot theory and its applications in dynamical systems, geometry, and topology, preprint (2015), arXiv:1501.05208v1] the second author constructed a connection between classical braid group and group presentation generated by elements corresponding to horizontal trisecants. This approach does not apply to links nor tangles because it requires that when counting trisecants, we have the same number of points at each level. For general tangles, trisecants passing through one component twice may occur. Free links can be obtained from tangles by attaching two end points of each component. We shall construct an invariant of free links and free tangles valued in groups as follows: we associate elements in the groups with 4-valent vertices of free tangles (or free links). For a free link with enumerated component, we “read” all the intersections when traversing a given component and write them as a group element. The problem of “pure crossings” of a component with itself by using the following statement: if two diagrams with no pure crossings are equivalent then they are equivalent by a sequence of moves where no intermediate diagram has a pure crossing. This statement is a result of a sort that an equivalence relation within a subset coincides with the equivalence relation induced from a larger set and it is interesting by itself.

In this paper, we recall the geometric definition of the braid group by Emil Artin and we give a complete, elementary geometric/topological proof of the standard presentation of the braid group on $n$ strings.

Let G be a group and h, g ∈ G. The 2-tuple (h, g) is said to be an n-Engel pair if h = [h,_n g] and g = [g,_n h]. In this paper, we will study the subgroup generated by the n-Engel pair under certain conditions.

Let G be a group and h, g ∈ G. The 2-tuple (h, g) is said to be an n-Engel pair, n ≥ 2, if h = [h,_n g], g = [g,_n h] and h ≠ 1. In this paper, we prove that if (h, g) is an n-Engel pair, hgh^-2gh = ghg and ghg^-2hg = hgh, then n = 2k where k = 4 or k ≥ 6. Furthermore, the subgroup generated by {h, g} is determined for k = 4, 6, 7 and 8.

Kim and Kostrikin constructed in [8, 9] a tessellation on the boundary of a polyhedral 3-cell consisting of 8n pentagons (n ≥ 1). The tessellation produces a family of closed connected orientable 3-manifolds (denoted by M₁(n) in the quoted papers) with spines corresponding to certain presentations of their fundamental groups. We investigate the topological and algebraic properties of such groups together with their derived quotients and split extensions, and completely classify the considered manifolds.

The purpose of this paper is to examine the relationship between groups, group presentations, their Cayley graphs, and associated Markov Processes. In particular, we prove that for a finite group derived from an ergodic Markov Process, a process we describe in the paper, the long range equilibrium vector is uniform on the group elements, as to be expected. We also prove a theorem giving a complete characterization of finitely generated free groups in terms of their associated Markov Processes.