One-dimensional actions of Higman's group

TL;DR

This paper constructs a faithful action of Higman's group on the real line using homeomorphisms, introduces many quasimorphisms, and proves that any smooth action by Higman's group on the line or circle is trivial.

Contribution

It provides the first known faithful action of Higman's group on the line and establishes the triviality of smooth actions, answering longstanding questions.

Findings

Constructed a faithful action of Higman's group on the line.

Generated numerous quasimorphisms from Higman's group to the reals.

Proved all $C^1$-diffeomorphism actions of Higman's group on the line or circle are trivial.

Abstract

We build a faithful action of Higman's group on the line by homeomorphisms, answering a question of Yves de Cornulier. As a by-product we obtain many quasimorphisms from the Higman group into the reals. We also show that every action by $C^1$-diffeomorphisms of Higman's group on the line or the circle is trivial.