On free discrete subgroups of Diff(I)

TL;DR

This paper demonstrates that the free group F_2 can be faithfully and discretely represented within the groups of orientation-preserving diffeomorphisms and homeomorphisms of the interval, exploring properties and raising questions about these representations.

Contribution

It establishes the existence of faithful discrete representations of F_2 into Diff_{+}(I) and Homeo_{+}(I), advancing understanding of group actions on the interval.

Findings

F_2 admits faithful discrete representations into Diff_{+}(I)

F_2 admits faithful discrete representations into Homeo_{+}(I)

Properties of these representations are analyzed

Abstract

We prove that a free group F_2 admits a faithful discrete representation into Diff_{+}(I). We also prove that F_2 admits a faithful discrete representation into Homeo_{+}(I). Some properties of these representations have been studied. In the last section we raise several questions.