Minimal presentation, finite quotients and lower central series of cactus groups

The cactus group $J_n$ first appeared in the works of Devadoss and Davis-Januszkiewicz-Scott under the name of quasibraid groups and mock reflection groups, respectively. These groups acts naturally on tensor products of objects of coboundary category, as braid groups act on braided monoidal category,  and especially on tensor products of crystals of  reductive Lie algebras of finite dimension.Moreover, as in braid groups, there is a diagrammatic representation of elements of cactus groups and a surjective morphism from $J_n$ to $S_n$, which kernel is the pure cactus groups $PJ_n$ which is the fundamental group of $\overline{M}^{n+1}_0(\mathbb{R})$, the Deligne-Knudson-Mumford moduli space of stable real curves of genus 0 with n + 1 marked points.The goal of this talk, based on a paper done with N. Nanda, is to introduce a new presentation for cactus groups and to use it to find some remarkable finite quotients of cactus groups. Then, this will allow us to compile some quotients of the lower central series of catus groups.