Tame Discrete Sets in Algebraic Groups

TL;DR

This paper extends the concept of tame discrete subsets from affine complex spaces to complex linear algebraic groups with trivial character groups, exploring their properties and generalizations.

Contribution

It generalizes the theory of tame discrete subsets from affine complex spaces to a broader class of algebraic groups with trivial character groups.

Findings

Extended the notion of tame discrete subsets to complex linear algebraic groups.

Analyzed properties of these subsets within the new context.

Provided foundational results for future research in algebraic group theory.

Abstract

Rosay and Rudin introduced the notion of Tame discrete subsets of the affine complex space and investigated their properties. We generalize this theory to the case of a complex linear algebraic group with trivial character group.

Full text

1. Introduction

For discrete subsets in ${\mathbb{C}}^{n}$ the notion of being “tame” was defined in the important paper of Rosay and Rudin [RR]. A discrete subset $D\subset{\mathbb{C}}^{n}$ is called “tame” if and only if there exists an automorphism $\phi$ of ${\mathbb{C}}^{n}$ such that $\phi(D)={\mathbb{N}}\times\{0\}^{n-1}$. (In this paper a subset $D$ of a topological space $X$ is called a “discrete subset” if every point $p$ in $X$ admits an open neighbourhood $W$ such that $W\cap D$ is finite.)

In [JW-TAME1] we introduced a new definition of tameness which applies to arbitrary complex manifolds and agrees with the tameness notion of Rosay and Rudin for the case $X\simeq{\mathbb{C}}^{n}$.

Definition 1.1**.**

*Let $X$ be a complex manifold. An infinite discrete subset $D$ is called (weakly) tame if for every exhaustion function $\rho:X\to{\mathbb{R}}^{+}$ and every map $\zeta:D\to{\mathbb{R}}^{+}$ there exists an automorphism $\phi$ of $X$ such that $\rho(\phi(x))\geq\zeta(x)$ for all $x\in D$. *

In terms of Nevanlinna theory a similar tameness notion may be formulated as follows: A discrete set is tame if its counting function $N(r,D)$ may be made as small as desired. (See §LABEL:sect-nevanlinna for details.)

First results about this notion obtained in [JW-TAME1] suggested that best results are to be expected in the case of complex manifolds whose automorphism group is very large in a certain sense. In particular, in [JW-TAME1] we proved some significant results for the case $X=SL_{n}({\mathbb{C}})$ and found some evidence suggesting that our tameness notion might not work very well for non-Stein manifolds like ${\mathbb{C}}^{n}\setminus\{(0,\ldots,0)\}$ or partially hyperbolic complex manifolds like $\Delta\times{\mathbb{C}}$.

In this paper we show that for complex manifolds biholomorphic to complex linear algebraic groups without non-trivial morphism to the multiplicative group ${\mathbb{C}}^{*}$ we obtain a theory of tame discrete sets essentially as strong as the theory which Rosay and Rudin developed for ${\mathbb{C}}^{n}$.

Andrist and Ugolini ([AU]) have proposed a different notion, namely the following: