Real elements in the mapping class group of $T^2$

TL;DR

This paper classifies elements of the torus's mapping class group that can be expressed as a product of two orientation-reversing involutions, linking algebraic properties with geometric structures like the Farey tessellation.

Contribution

It provides a complete classification of such elements in the mapping class group of the torus, connecting algebraic decompositions with geometric and topological features.

Findings

Classification of elements with involution decompositions

Connection to monodromy maps of real fibrations

Use of Farey tessellation and $SL(2,\

Abstract

We present a complete classification of elements in the mapping class group of the torus which have a representative that can be written as a product of two orientation reversing involutions. Our interest in such decompositions is motivated by features of the monodromy maps of real fibrations. We employ the property that the mapping class group of the torus is identifiable with $SL(2,\Z)$ as well as that the quotient group $PSL(2,\Z)$ is the symmetry group of the {\em Farey tessellation} of the Poincar\'e disk.