Sakasai - On groups of Kim-Manturov

Contents

We look at the group ${\Gamma }_n^4$ (from Kim-Manturov ‘19) The presentation of the group is described.

The group can be defined by triangulations of polyhedrons. The group elements move between the triangulations. (see “Invariants and Pictures” by manturov)

Tadokoro gave a few combinatorial properties of the Kim Manturov Groups.

New results

The group ${\Gamma }_5^4$ has a subgroup of index $2$. We know the presentation of this group.

Related groups

We define groups which are similar. (like an increasing-order version). This group is nice since it has a pretty easy presentation.

Groups

• Is it a non-commutative, infinite group for $n\ge 6$
• Is ${\Gamma }_n^4\to {\Gamma }_{n+1}^4$
• Is the Word problem solved?
• Is the natural homomorphism of the similar group ${\Delta }_n\to {\Gamma }_n^4$ injective?
• They might be coxeter
• Any good spaces the groups act on?