Injections of mapping class groups

TL;DR

This paper introduces new injective homomorphisms between mapping class groups of surfaces, revealing surprising behaviors such as pseudo-Anosov elements mapping to multi-twists, expanding understanding of surface symmetries.

Contribution

It constructs novel monomorphisms between different mapping class groups, including unexpected mappings of pseudo-Anosov elements, which were not previously known.

Findings

Injective maps from one surface's mapping class group to another's.

Pseudo-Anosov elements can map to multi-twists under these injections.

Elementary constructions produce phenomena previously thought impossible.

Abstract

We construct new monomorphisms between mapping class groups of surfaces. The first family of examples injects the mapping class group of a closed surface into that of a different closed surface. The second family of examples are defined on mapping class groups of once-punctured surfaces and have quite curious behaviour. For instance, some pseudo-Anosov elements are mapped to multi-twists. Neither of these two types of phenomena were previously known to be possible although the constructions are elementary.