Test Map and Discreteness Criteria for Subgroups in PU(1,n;C)

TL;DR

This paper introduces a new method using a test map to determine the discreteness of subgroups in PU(1,n;C), and establishes conditions under which subgroup discreteness can be inferred from two-generator subgroups.

Contribution

It presents a novel approach for testing subgroup discreteness via a test map not necessarily in G and links subgroup discreteness to two-generator subgroup discreteness.

Findings

Test map can be used to determine subgroup discreteness.

Discreteness of all two-generator subgroups implies the entire subgroup is discrete.

Provides criteria for subgroup discreteness in PU(1,n;C).

Abstract

We study the discreteness for non-elementary subgroup G in PU(1, n;C), under the assumption that G satisfies Condition A. Mainly, we present that one can use a test map, which need not to be in G, to examine the discreteness of G, and also show that G is discrete, if every two-loxodromic-generator subgroup of G is discrete.