The Eudoxus reals is a "non-conventional" construction of reals, that bypasses any construction of the rationals, instead progressing directly from $\mathbb{Z}$ to $\mathbb{R}$ (or in some versions, from $\mathbb{N}$ to $\mathbb{R}$, adding negatives only later). A good explanation can be found in The Eudoxus Real Numbers by Arthan. To accomplish this, it uses the notion of "almost homomorphisms" from $f : \mathbb{Z} \to \mathbb{Z}$, that is $f$ such that $\{f(x+y) - f(x) - f(y)\}$ is bounded, or equivalently, finite. These $f$ and $g$ correspond to reals where addition of reals is pointwise addition of functions, and multiplication of reals is composition.
Only at the very end does of the linked paper does Arthan mention the generalization of this construction to alternate Abelian groups, as a functor $\mathbb{E}(G,H)$. I am somewhat suspicious of this notation, as this will not in general allow a notion of composition (and thus a notion of 'multiplication') unless $G = H$, so I will write simply $\mathbb{E}(G)$ so that we get both operations.
I am curious if there are any studies to the behavior when $G$ is no longer Abelian. At least for a few simple, small cases (e.g. the infinite dihedral group) I was able to convince myself that it wasn't too interesting (in that case, it produced the reals again). This is hampered by the fact that if the group is actually finite then the construction collapses.
The boundedness requirement would now refer to the set $\{f(x^{-1})f(xy)f(y^{-1})\}$, or alternately $\{f(x)^{-1}f(xy)f(y)^{-1}\}$.
But I would be curious what happens if e.g. the infinite symmetric group (consisting of permutations on $\mathbb{N}$ transposing only finitely many elements) is used. One can use "finite" as one's meaning for "bounded" set, but I feel that "bounded in word length with regards to some set of generators" may be better. But then, that would depend on the set of generators chosen. (I can think of at least two natural ones that aren't immediately equivalent)
To be precise, I tried computing some almost homomorphisms with regards to actual finite sets, or elements of finite length with regards to transpositions, but got nowhere.
Is anyone aware of literature on this? Or capable of producing a proof that for 'very nonabelian' groups like this (groups with no subquotients of $\mathbb{Z}$?) that there are non-trivial almost homomorphisms? I would also be curious to see whether multiplication is still closed and whether addition is still
