Tame discrete subsets in Stein manifolds

TL;DR

This paper generalizes the concept of 'tame' discrete subsets from complex Euclidean spaces to complex manifolds, particularly semisimple Lie groups, and demonstrates that classical results extend to these broader contexts.

Contribution

It introduces a new, general definition of tameness for complex manifolds and extends classical results from ${f C}^n$ to semisimple complex Lie groups.

Findings

Tameness can be defined for arbitrary complex manifolds.

Classical results for ${f C}^n$ extend to semisimple complex Lie groups.

Permutations of $SL(2,{f Z})$ extend to biholomorphic maps of $SL(2,{f C})$.

Abstract

For discrete subsets in ${\bf C}^n$ the notion of being "tame" was defined by Rosay and Rudin. We propose a general definition of "tameness" for arbitrary complex manifolds and show that many results classically known for ${\bf C}^n$ may be generalized to semisimple complex Lie groups. For example, every permutation of $SL(2,{\bf Z})$ extends to a biholomorphic self-map of $SL(2,{\bf C}$.