In this paper, we define the notion of a virtually symmetric representation representations virtual braid groups and prove that many known are equivalent to symmetric. Using one such representation, link which is an extension defined by Kauffman. Moreover, introduce concept marked Gauss diagrams as generalisation their interpretation in terms knot-like diagrams. We extend definition peripheral ...

Let M be a graph manifold. We prove that fundamental groups of embedded incompressible surfaces in M are separable in π1M , and that the double cosets for crossing surfaces are also separable. We deduce that if there is a ‘sufficient’ collection of surfaces in M , then π1M is virtually the fundamental group of a special non-positively curved cube complex. We provide a sufficient collection for ...

The aim of this paper is to prove Theorem 1.1 below, a generalization to virtually nilpotent spaces of a result of Wilkerson [13] and Sullivan [12]. We recall that a CW complex Y is virtually nilpotent if (i) Y is connected, (ii) 7r~ Y is virtually nilpotent (i.e. has a nilpotent subgroup of finite index) and (iii) for every integer n > 1, zr~Y has a subgroup of finite index which acts nilpoten...

Over the past decades, numerous twin studies have revealed moderate to high heritability estimates for individual differences in a wide range of human traits, including cognitive ability, psychiatric disorders, and personality traits. Even factors that are generally believed to be environmental in nature have been shown to be under genetic control, albeit modest. Is such heritability also prese...

LetM be a compact oriented irreducible 3–manifold which is neither a graph manifold nor a hyperbolic manifold. We prove that π1M is virtually special.