Let $TB_{n}$ be the group defined by the presentation with generators $t_{1},...,t_{n-2}$ and relations $t_{i}t_{i+1}t_{i+2}t_{i}=t_{i+2}t_{i}t_{i+1}t_{i+2}$ and $t_{i}t_{j}=t_{j}t_{i}$ whenever $|i-j|>2$. We shall call $TB_{n}$ the ternary braid group. The group $TB_{n}$ has several of the features of the braid group $B_{n}$. For example, there is an injective group homomorphism $\phi:B_{n}\rightarrow TB_{2n}$ defined by $\phi(\sigma_{i})=t_{2i-1}t_{2i}$. Then group $TB_{n}$ can also be endowed with a ternary self-distributive operation similar to shifted conjugacy. The group $TB_{n}$ even acts on ternary racks and quandles.
In addition to $TB_{n}$, there are also several topological ways of generalizing the braid group to various groups. For instance, the braid groups appear as the mapping class groups and fundamental groups of certain topological spaces. One can also consider higher dimensional braids. Is $TB_{n}$ isomorphic to any higher dimensional or generalized braid group? Does there exist a higher dimensional or generalized braid group $G$ along with a homomorphism $\phi:TB_{n}\rightarrow G$ of small kernel or a homomorphism $\theta:G\rightarrow TB_{n}$ with small kernel? Informally, does the group $TB_{n}$ nearly arise in algebraic topology?
