Presentations, embeddings and automorphisms of homogeneous spaces for $\mathrm{SL}_2(k)$

Let $k$ be an algebraically closed field of characteristic zero and $G$ be a linear algebraic $k$-group. It is well known that every affine $G$-variety admits a $G$-equivariant closed embedding into a finite-dimensional $G$-module. Such an embedding is a presentation of the $G$-variety, and a presentation is called minimal if the dimension of the corresponding $G$-module is minimal. The problem of finding a minimal presentation generalizes the problem of determining whether a group action on affine space is linearizable.
We discuss a minimal presentation for each homogeneous space of $\mathrm{SL}_2(k)$. Of particular interest are the surfaces $Y = \mathrm{SL2(k)/T$ and $X = \mathrm{SL}_2(k)/N$, where $T$ is the one-dimensional torus and $N$ is its normalizer.
In the previous talk, it was shown that the minimal presentation of $X$ has dimension $5$, while the embedding dimension of $X$ is $4$, and there exists no closed $\mathrm{SL}_2(k)$-equivariant embedding of $X$ into $\mathbb{A}^4$. Thus, the $\mathrm{SL}_2(k)$-action on $X$ is absolutely nonextendable to $\mathbb{A}^4$. In addition, $X$ is noncancelative, that is, there exists a surface $X'$ such that $X \times \mathbb{A}^1$ is isomorphic to $X \times \mathbb{A}^1$ and $X$ is not isomorphic to $X'$. We consider two other examples of surfaces with absolutely nonextendable group actions.
This talk is based on the work [Gene Freudenburg. Presentations, embeddings and automorphisms of homogeneous spaces for $\mathrm{SL}_2(\mathbb{C})$. arXiv:2504.21712].