A new instance of the Lück approximation in the positive characteristic

Let $\Gamma$ be a group and let $K$ be a field. For every matrix $A\in \text{Mat}_{n\times m}(K[\Gamma])$ and every normal subgroup $N$ of $\Gamma$ of finite index let us define $$\begin{array}{cccc} \phi_{\Gamma/N}^A: & K[\Gamma/N]^n & \to & K[\Gamma/N]^m \\ &(x_1,\ldots, x_n)&\mapsto & (x_1,\ldots, x_n)A\end{array}.$$ This is a $K$-linear map between two finite-dimensional $K$-vector spaces. Thus, we can define a Sylvester rank function of $K[\Gamma]$ by means of $$\text{rk}_{\Gamma/N}(A)=\frac{\text{dim}_K \text{Im} \phi_{\Gamma/N}^A}{|\Gamma:N|}.$$

Conjecture [The Lück approximation conjecture] Let $\Gamma$ be a group, $K$ a field and $\Gamma>N_1>N_2>\ldots $ a descending chain of normal subgroups of $\Gamma$ of finite index with trivial intersection. Let $A$ be a matrix over $K[\Gamma]$. Then the following holds:

1. The sequence $\{\text{rk}_{\Gamma/N_i}(A)\}_{i\ge 1}$ converges.
2. The limit $\displaystyle \lim_{i\to \infty} \text{rk}_{\Gamma/N_i}(A)$ does not depend on the chain $\Gamma>N_1>N_2>\ldots $.
3. If moreover $\Gamma$ is locally indicable, then there exists an embedding $K[\Gamma]\hookrightarrow \mathcal Q$ into a division ring $\mathcal Q$ such that $\displaystyle \lim_{i\to \infty} \text{rk}_{\Gamma/N_i}(A)=\text{rk}_{\mathcal Q}(A).$

All three statements were proven if $K$ has characteristic 0. However, if $K$ has positive characteristic, in full generality, (1), (2), and (3) are only known when $\Gamma$ is amenable. In my talk, I will show that the conjecture holds in the case where $K=\mathbb F_p$ and the inverse limit $\displaystyle \varprojlim_{i\in \mathbb N} \Gamma/N_i$ is a free-by-$\mathbb Z_p$ pro-$p$ group. This is a joint work with Henrique Souza arXiv:2402.14130