The unrestricted virtual braid groups UV Bn

Abstract

Let [Formula: see text] and [Formula: see text] be the unrestricted virtual braid group and the unrestricted virtual pure braid group on n strands, respectively. We study the groups [Formula: see text] and [Formula: see text], and our main results are as follows: for [Formula: see text], we give a complete description, up to conjugation, to all possible homomorphisms from [Formula: see text] to the symmetric group [Formula: see text]. For [Formula: see text], we characterize all possible images of [Formula: see text], under a group homomorphism, to any finite group [Formula: see text]. For [Formula: see text], we prove that [Formula: see text] is a characteristic subgroup of [Formula: see text]. In addition, we determine the automorphism group of [Formula: see text] and we prove that [Formula: see text] is a subgroup of the outer automorphism group of [Formula: see text]. Lastly, we show that [Formula: see text] and [Formula: see text] are residually finite and Hopfian but not co-Hopfian. We also remark that some of these results hold accordingly for the welded braid group [Formula: see text] and we discuss about its automorphism group.