On the structure of groups defined by Kim and Manturov

We consider a series of groups $\Gamma_n^4$ defined by Kim and Manturov. These groups have their background in Delaunay triangulations of a plane and they are expected to have relationships to many geometric objects. In this talk, by a group theoretical argument, we show that the groups $\Gamma_n^4$ are finite for all n $\ge 6$ and in fact they are 2-step nilpotent 2-groups. This is a joint work with Carl-Fredrik Nyberg-Brodda, Yuuki Tadokoro and Kokoro Tanaka (arXiv: 2506.05778, 2506.08050).