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We construct a group $\Gamma_{n}^{4}$ corresponding to the motion of pointsin $\mathbb{R}^{3}$ from the point of view of Delaunay triangulations. We studyhomomorphisms from pure braids on $n$ strands to the product of copies of$\Gamma_{n}^{4}$. We will also study the group of pure braids in$\mathbb{R}^{3}$, which is described by a fundamental group of the restrictedconfiguration space of $\mathbb{R}^{3}$, and define the group homomorphism fromthe group of pure braids in $\mathbb{R}^{3}$ to $\Gamma_{n}^{4}$. In the end ofthis paper we give some comments about relations between the restrictedconfiguration space of $\mathbb{R}^{3}$ and triangulations of the 3-dimensionalball and Pachner moves.

Artin's braids, Braids for three space, and groups $\Gamma_{n}^{4}$ and $G_{n}^{k}$