A weak Zassenhaus lemma for discrete subgroups of Diff(I)

TL;DR

This paper establishes a weaker form of the Zassenhaus Lemma for subgroups of Diff(I) and explores conditions under which certain groups do not admit discrete faithful representations.

Contribution

It introduces a weaker Zassenhaus Lemma for Diff(I) subgroups and characterizes groups that lack discrete faithful representations based on their commutator subgroup.

Findings

Proved a weaker Zassenhaus Lemma for Diff(I)

Identified groups with free subsemigroup in the commutator do not admit C_0-discrete faithful representations

Extended understanding of discreteness in groups of diffeomorphisms

Abstract

We prove a weaker version of Zassenhaus Lemma (also known as Margulis Lemma) for subgroups of Diff(I). We also show that a group with commutator subgroup containing a free subsemigroup does not admit a C_0-discrete faithful representation in Diff(I).