The generalized Dehn twist along a figure eight

TL;DR

This paper introduces a generalized Dehn twist for loops with a single transverse double point on surfaces and demonstrates that such twists are not realizable as mapping classes, extending understanding of surface automorphisms.

Contribution

It defines generalized Dehn twists along figure-eight loops and proves their non-realizability as mapping classes, broadening the scope of surface automorphism studies.

Findings

Generalized Dehn twists are defined for loops with double points.

Such twists are not realizable as mapping classes.

The work extends the understanding of automorphisms of surface groups.

Abstract

For any unoriented loop on a compact connected oriented surface with one boundary component, the generalized Dehn twist along the loop is defined as an automorphism of the completed group ring of the fundamental group of the surface. If the loop is simple, this is the usual right handed Dehn twist, in particular realized as a mapping class of the surface. We investigate the case when the loop has a single transverse double point, and show that in this case the generalized Dehn twist is not realized as a mapping class.