Automorphism groups of affine varieties without non-algebraic elements

Given an affine algebraic variety $X$, we study when the neutral component $\mathrm{Aut}^0(X)$ of the automorphism group consists of algebraic elements. It is conjectured that the following conditions on $\mathrm{Aut}^0(X)$ are equivalent:
- all unipotent elements (hence all $\mathbb{G}_a$-actions on $X$) commute,
- it consists of algebraic elements,
- it is nested, i.e., a direct limit of algebraic subgroups,
- it is a semidirect product of an algebraic torus and an abelian unipotent group.

Earlier we proved the conjecture for the group generated by connected algebraic subgroups instead of $\mathrm{Aut}^0(X)$. In this talk we present our further development: we proved that $\mathrm{Aut}^0(X)$ consists of algebraic elements if and only if it is nested. To prove it, we obtained the following fact: if a connected ind-group $G$ contains a closed connected ind-subgroup $H \subset G$ with a geometrically smooth point, and for any $g \in G$ some power of $g$ belongs to $H$, then $G=H$.

We will also discuss possible approaches to the conjecture and related questions. The talk is based on the joint work with Andriy Regeta.