Unicellular maps and filtrations of the mapping class group

TL;DR

This paper explores the structure of unicellular maps and their relationship with the mapping class group, revealing new filtrations and subgroup encodings that enhance understanding of their interconnected topology.

Contribution

It introduces a novel approach to describe the mapping class group via unicellular maps and provides new filtrations and subgroup encodings.

Findings

Connectedness of new graph families on unicellular maps established

Mapping class group described through surgeries on unicellular maps

Filtrations of the mapping class group constructed

Abstract

This article first answers to questions about connectedness of a new family of graphs on unicellular maps. Answering these questions goes through a description of the mapping class group as surgeries on unicellular maps. We also show how unicellular maps encode subgroups of the mapping group and provide filtrations of the mapping class group. These facts add a layer on the ubiquitous character of unicellular maps.