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Let VB$_n$ be the virtual braid group on $n$ strands and let $mathfrak{S}_n$ be the symmetric group on $n$ letters. Let $n,m in mathbb{N}$ such that $n ge 5$, $m ge 2$ and $n ge m$. We determine all possible homomorphisms from VB$_n$ to $mathfrak{S}_m$, from $mathfrak{S}_n$ to VB$_m$ and from VB$_n$ to VB$_m$. As corollaries we get that Out(VB$_n$) is isomorphic to $mathbb{Z}/2mathbb{Z} times mathbb{Z}/2mathbb{Z}$ and that VB$_n$ is both Hopfian and co-Hofpian.
We prove that finitely generated virtually free groups are stable in permutations. As an application, we show that almost-periodic almost-automorphisms of labelled graphs are close to periodic automorphisms.
We use wreath products to provide criteria for a group to be conjugacy separable or omnipotent. These criteria are in terms of virtual retractions onto cyclic subgroups. We give two applications: a straightforward topological proof of the theorem of
On March 2004, Anshel, Anshel, Goldfeld, and Lemieux introduced the emph{Algebraic Eraser} scheme for key agreement over an insecure channel, using a novel hybrid of infinite and finite noncommutative groups. They also introduced the emph{Colored Bur
We show that word hyperbolicity of automorphism groups of graph products $G_Gamma$ and of Coxeter groups $W_Gamma$ depends strongly on the shape of the defining graph $Gamma$. We also characterized those $Aut(G_Gamma)$ and $Aut(W_Gamma)$ in terms of $Gamma$ that are virtually free.
We show that for every finitely generated closed subgroup $K$ of a non-solvable Demushkin group $G$, there exists an open subgroup $U$ of $G$ containing $K$, and a continuous homomorphism $tau colon U to K$ satisfying $tau(k) = k$ for every $k in K$.