The n-simplex equation (n-SE) was introduced by A. B. Zamolodchikov as a generalization of the Yang-Baxter equation, which is the 2-simplex equation in these terms. In the present paper we suggest some general approaches to constructing solutions of n-simplex equations, describe some types of solutions, introduce an operation which under some conditions allows us to construct a solution of (n + m + k)-SE from solution of (n + k)-SE and (m + k)-SE. We consider the tropicalization of rational solutions and discuss a way to generalize it. We prove that if a group G is an extension of a group H by a group K, then we can find a solution of the n-SE on G from solutions of this equation on H and on K. Also, we find solutions of the parametric Yang-Baxter equation on H with parameters in K. For studying solutions of the 3-simplex equations we introduce algebraic systems with ternary operations and give examples of these systems which gives solutions of the 3-SE. We find all elementary verbal solutions of the 3-SE on free groups.

In the present paper we study structural aspects of certain quotients of braid groups and virtual braid groups. In particular, we construct and study linear representationswhich are connected with the famous Lawrence-Bigelow-Krammer representation. It turns out that these representations are faithful representations of crystallographic groups Bn/P ′ n , V Bn/V P ′ n , respectively. Using these representations we study certain properties of the groups Bn/P ′ n , V Bn/V P ′ n . Moreover, we construct new representations and decompositions of universal braid groups U Bn.