We investigate the cohomology of level 4 subgroup braid group, namely, kernel mod reduction Burau representation at $$t=-1$$ . This group is also equal to 2 abelianization pure group. give an exact formula for first Betti number; it a quartic polynomial in number strands. show that, like homology satisfies uniform stability sense Church and Farb. Unlike symmetries—the quotient by subgroup—is on...

By a quasi-permutation matrix we mean a square matrix over the complex field C with non-negative integral trace. Thus, every permutation matrix over C is a quasipermutation matrix. For a given finite group G, let p(G) denote the minimal degree of a faithful permutation representation of G (or of a faithful representation of G by permutation matrices), let q(G) denote the minimal degree of a fa...

the probability that the commutator of two group elements is equal to a given element has been introduced in literature few years ago. several authors have investigated this notion with methods of the representation theory and with combinatorial techniques. here we illustrate that a wider context may be considered and show some structural restrictions on the group.

First, we apply Thurston’s construction of pseudo-Anosov homeomorphisms to grid graphs and obtain translation surfaces whose Veech groups are commensurable to (m, n,∞) triangle groups. Many (if not all) of the surfaces we construct were first discovered by Bouw and Möller, however our treatment of the surfaces differs. We construct these surfaces by gluing together polygons in two ways. We use ...

Abstract. We propose the fundamental and two dimensional representation of the Lorentz groups on a (3+1)-dimensional hypercubic lattice, from which representations of higher dimensions can be constructed. For the unitary representation of the discrete translation group we use the kernel of the Fourier transform. From the Dirac representation of the Lorentz group (including reflections) we deriv...