Short proofs of theorems of Malyutin and Margulis

TL;DR

This paper provides a concise proof of Malyutin's theorem on minimal circle actions and uses it to confirm Margulis' theorem, supporting the Ghys-Margulis alternative through the application of enveloping semigroup techniques.

Contribution

It introduces a shorter proof of Malyutin's theorem and derives Margulis' theorem, simplifying the understanding of the Ghys-Margulis alternative.

Findings

Short proof of Malyutin's theorem

Confirmation of Margulis' theorem

Support for the Ghys-Margulis alternative

Abstract

The Ghys-Margulis alternative asserts that a subgroup $G$ of homeomorphisms of the circle which does not contain a free subgroup on two generators must admit an invariant probability measure. Malyutin's theorem classifies minimal actions of $G$. We present a short proof of Malyutin's theorem and then deduce Margulis' theorem which confirms the G-M alternative. The basic ideas are borrowed from the original work of Malyutin but the use of the apparatus of the enveloping semigroup enables us to shorten the proof considerably.