Combinatorics of affine cactus groups

TL;DR

This paper explores the combinatorial structure of affine cactus groups, showing their relation to Coxeter groups and embedding properties, which lead to new insights into their algebraic characteristics.

Contribution

It introduces a description of affine cactus groups as cactus groups on Coxeter groups of type eAn and proves their embedding into a semi-direct product, revealing new algebraic properties.

Findings

Affine cactus groups can be described as cactus groups on Coxeter groups of type eAn.

These groups embed into a semi-direct product of Coxeter groups.

The groups have a trivial center and solvable word problem.

Abstract

This article deals with the study of affine cactus groups from a combinatorial point of view. Those groups are extensions of cactus groups, which are related to braid and diagram groups and have gained an important place in many mathematics topics. We first show that affine cactus groups may be described as cactus groups on Coxeter groups of type eAn. Then, we prove that these groups embed into a semi-direct product of Coxeter groups, which allows us to obtain a number of combinatorial properties of affine cactus groups, such as the solubility of the world problem or the fact that their centre is trivial.