I have heard recently that one can define the usual braid groups $B_n$ using almost actions (in the sense of Yves Cornulier, as in https://arxiv.org/pdf/1901.05065) using the symmetric groups $S_n$. I remember the construction in rough details, but I am struggling to come up with precise statements. I would be very pleased to see a working construction or a reference.
For this question to be self-contained, I will explain the definition for almost actions and the rough idea that I have. For sets $X, Y$, an almost-bijection from $X$ to $Y$ is a bijection from a cofinite subset of $X$ to a cofinite subset of $Y$. Naturally, one can define compositions and inverses for almost-bijections. So, as usual, an almost action of a group $G$ on a set $X$ is just a presentation $G \to AlmostBijections(X, X)$.
The construction that I heard for the braid group $B_n$ goes like this: one takes X tobe either $X = [n] \times \mathbb{N}$, where $[n]$ is an $n$ element set and $\mathbb{N}$ are natural numbers, or to be $X = [n] \times \mathbb{Z}$.
For the first set $X$ I heard that construction goes like this: one generates a subgroup of $AlmostBijections([n] \times \mathbb{N})$ using the following $n-1$ almost actions $s_i (t, m) = ((i, i+1)\cdot t, m+1)$ with $i= 0...n-1$, here $(i, i+1)\cdot t$ is the usual action of transposition $(i, i+1)$ on the first coordinate $t \in [n]$. Ideologically, these $s_i$ should serve as the usual Artin generators for $B_n$. However this definition does not work, since $s_i^2$ would map $(t, m) \mapsto (t, m+2)$ regardless of $i$.