Free dense subgroups of holomorphic automorphisms

TL;DR

This paper demonstrates the existence of free dense subgroups generated by two elements within the holomorphic automorphism groups of complex spaces and Stein manifolds, highlighting the role of generalized translations and hypercyclic conjugation operators.

Contribution

It extends the construction of free dense subgroups to broader classes of complex manifolds using generalized translations and hypercyclicity techniques.

Findings

Existence of free dense subgroups in holomorphic shear and overshear groups.

Extension of results to Stein manifolds with the Density Property.

Hypercyclicity of conjugation operators related to generalized translations.

Abstract

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.