Holomorphic automorphisms of Danielewski surfaces II -- structure of the overshear group

Holomorphic Lie Group Actions on Danielewski Surfaces

We prove that any Lie subgroup $G$ (with finitely many connected components) of an infinite-dimensional topological group $\mathcal G$ which is an amalgamated product of two closed subgroups, can be conjugated to one factor. We apply this result to classify Lie group actions on Danielewski surfaces by elements of the overshear group (up to conjugation).

代数过剪密度性质

Algebraic Overshear Density Property

本文提出代数过剪密度性质，统一了代数灵活性与全纯密度性质，并证明具有该性质的复仿射曲面中带边黎曼面存在真全纯嵌入。

Free dense subgroups of holomorphic automorphisms

We show the existence of free dense subgroups, generated by 2 elements, in the holomorphic shear and overshear group of complex-Euklidean space and extend this result to the group of holomorphic automorphisms of Stein manifolds with Density Property, provided there exists a generalized translation. The conjugation operator associated to this generalized translation is hypercyclic on the topological space of holomorphic automorphisms.

Manifolds with infinite dimensional group of holomorphic automorphisms and the Linearization Problem

We overview a number of precise notions for a holomorphic automorphism group to be big together with their implications, in particular we give an exposition of the notions of flexibility and of density property. These studies have their origin in the famous result of Andersén and Lempert from 1992 proving that the overshears generate a dense subgroup in the holomorphic automorphism group of $\C^n, n\ge 2$. There are many applications to natural geometric questions in complex geometry, several of which we mention here. Also the Linearization Problem, well known since the 1950 s and considered b…

Lie algebra generated by locally nilpotent derivations on Danielewski surfaces

We give a full description of the Lie algebra generated by locally nilpotent derivations (short LNDs) on smooth Danielewski surfaces $D_p$ given by $xy=p(z)$. In case $\mathrm{deg}(p)\geq 3$ it turns out to be not the whole Lie algebra $\mathrm{VF}_{alg}^ω(D_p)$ of volume preserving algebraic vector fields, thus answering a question posed by Lind and the first author. Also we show algebraic volume density property (short AVDP) for a certain homology plane, a homogeneous space of the form $SL_2 (\mathbb{C}) /N$, where $N$ is the normalizer of the maximal torus and another related example. At th…