In the 1940's, Ulam asked the following question, now known as "Ulam's stability problem": Given an approximate homomorphism between a group $\Gamma$ and a metric group $(G,d)$, how close is it to an actual homomorphism?

The answer depends on the choices of $\Gamma$ and $(G,d)$ as well as one's definition of approximate homomorphism and distance between functions.

In this talk we are going to focus on the case where:

• $G$ is the symmetric group $Sym(n)$ endowed with the normalized Hamming metric $d_H$.

• A function $f: \Gamma \to Sym(n)$ is an $\epsilon$-approximate homomorphism if for every $x,y\in \Gamma$ we have $d_H(f(xy),f(x)f(y))\leq \epsilon$.

• The distance between functions is the $L^\infty$ one.

Let $\Gamma=SL_r(Z)$ where $r\geq 3$. The plan for the talk is as follows:

1. For every $\epsilon>0$ we can find an $\epsilon$-approximate homomorphism from $SL_r(Z)$ to some $Sym(n)$ such that it is at least $1/2$ away from any genuine homomorphism to the same Sym(n). This will motivate us to define a flexible notion of stability, similar to the one initiated by Becker--Lubotzky and Gowers--Hatami.

2. We will then show, using a method invented by Burger--Ozawa--Thom, that if one is proving a flexible stability result for all amenable groups, then using bounded generation one can deduce the same for $SL_r(Z)$ when $r\geq 3$.

3. Lastly, we will outline the proof that every amenable group is flexibly stable (in the sense of this talk).

This is based on joint work with Oren Becker.