
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.
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Tame Discrete Sets in Algebraic Groups
Jörg Winkelmann
Jörg Winkelmann
Lehrstuhl Analysis II
Mathematisches Institut
Ruhr-Universität Bochum
44780 Bochum
Germany
ORCID: 0000-0002-1781-5842 ](mailto:[email protected]%0A)
Key words and phrases:
tame discrete sets, linear algebraic group, Stein manifold
1991 Mathematics Subject Classification:
32M17
1. Introduction
For discrete subsets in the notion of being “tame” was defined in the important paper of Rosay and Rudin [RR]. A discrete subset is called “tame” if and only if there exists an automorphism of such that . (In this paper a subset of a topological space is called a “discrete subset” if every point in admits an open neighbourhood such that 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 .
Definition 1.1**.**
*Let be a complex manifold. An infinite discrete subset is called (weakly) tame if for every exhaustion function and every map there exists an automorphism of such that for all . *
In terms of Nevanlinna theory a similar tameness notion may be formulated as follows: A discrete set is tame if its counting function 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 and found some evidence suggesting that our tameness notion might not work very well for non-Stein manifolds like or partially hyperbolic complex manifolds like .
In this paper we show that for complex manifolds biholomorphic to complex linear algebraic groups without non-trivial morphism to the multiplicative group we obtain a theory of tame discrete sets essentially as strong as the theory which Rosay and Rudin developed for .
Andrist and Ugolini ([AU]) have proposed a different notion, namely the following:
