Group stability considers the question of whether "almost"-homomorphisms are "close to" true homomorphisms. Here, "almost" and "close to" are made rigorous using a group metric.
More precisely, suppose we have a group $\Gamma$, and a family $\{G_n\}$ of groups equipped with a bi-invariant metric $d_n$. An almost (or asymptotic) homomorphism from $\Gamma$ to the family $\{G_n\}$ is a sequence of maps $$\phi_n: \Gamma \to G_n$$ such that for all $x,y \in \Gamma$, $$d_n\left( \phi_n(xy), \phi_n(x)\phi_n(y) \right) \longrightarrow 0$$ as $n \to \infty$. The group $\Gamma$ is said to be stable with respect to $\{G_n\}$ if any such almost homomorphism is close to a sequence of homomorphisms. That is, given any almost homomorphism $\{\phi_n\}$, there exists a sequence $\{\psi_n:\Gamma \to G_n\}$ of homomorphisms such that for all $x \in \Gamma$, $$d_n\left(\psi_n(x),\phi_n(x) \right) \longrightarrow 0$$ as $n \to \infty$.
This phenomenon of stability can also be interpreted as a lifting problem by combining all the groups $G_n$ into a direct product. Consider the direct product $\mathcal{G}=G_1 \times G_2 \times \dots$ and let $N$ be its subset of elements $(g_1,g_2,\dots)$ such that $d_n(g_n, 1) \to 0$ as $n \to \infty$ (that is, $N$ is the subset of elements (or sequences) that converge to the identity element). This $N$ is a normal subgroup of $\mathcal{G}$ , and an almost homomorphism $\{\phi_n\}$ can be interpreted simply as a homomorphism $$\phi: \Gamma \to \mathcal{G}/N$$ The stability question is now whether this homomorphism $\phi$ from $\Gamma$ to the quotient $\mathcal{G}/N$, can be lifted to a homomorphism $\psi: \Gamma \to \mathcal{G}$.
Note that the speed of convergence was never an issue above. What if we consider uniform stability? Now we have an almost homomorphism $$\phi_n: \Gamma \to G_n$$ such that given any $\epsilon>0$, then there exists $N$ such that for all $x,y \in \Gamma$, $$d_n\left( \phi_n(xy), \phi_n(x)\phi_n(y) \right) < \epsilon$$ when $n \geq N$. That is, the rate of convergence to $0$ is uniform over all elements of $\Gamma$. The group $\Gamma$ is said to be uniformly-stable with respect to $\{G_n\}$ if for any such uniformly almost homomorphism, there exists a sequence $\{\psi_n:\Gamma \to G_n\}$ of homomorphisms such that given any $\epsilon>0$, there exists $N$ such that for all $x \in \Gamma$, $$d_n\left(\psi_n(x),\phi_n(x) \right) < \epsilon$$ for $n \geq N$.
Do we have an interpretation of uniform stability in terms of lifting homomorphisms from a quotient group as in the non-uniform case?
The main difficulty is that when quotienting out $N$ from $\mathcal{G}$, the information regarding uniformity is lost. If instead of taking the homomorphism $$\phi: \Gamma \to \mathcal{G}/N$$ we take the homomorphism to be $$\phi: \Gamma \times \Gamma \times \dots \to \mathcal{G}/N$$ would this work?
Relevant References: 1) A survey by Andreas Thom on group approximations: https://arxiv.org/abs/1712.01052
2) Notes/thesis of Christoph Gamm on uniform stability https://www.math.uni-leipzig.de/preprints/p1110.0020.pdf
