Elements of Torelli topology: I. The rank of abelian subgroups

TL;DR

This paper introduces a new, clearer approach to understanding abelian subgroups in Torelli groups, providing stronger estimates and a novel, shorter proof for characterizing certain Dehn twists, advancing Torelli topology research.

Contribution

It offers new proofs and estimates for abelian subgroups in Torelli groups, and presents a shorter, more transparent proof for algebraic characterizations of specific Dehn twists.

Findings

Stronger bounds on the rank of abelian subgroups in Torelli groups.

A shorter proof for the algebraic characterization of certain Dehn twists.

Insights into the extension problem in Torelli groups, setting the stage for further research.

Abstract

By Torelli topology the author understands aspects of the topology of surfaces (potentially) relevant to the study of Torelli groups. The present paper is devoted to a new approach to the results of W. Vautaw about Dehn multi-twists in Torelli groups and abelian subgroups of Torelli groups. The new proofs are more transparent and lead to stronger estimates of the rank of an abelian subgroup when some additional information is available. An unexpected application of our methods is a new proof of the algebraic characterization of the Dehn twist about separating circles and the Dehn-Johnson twists about bounding pairs of circles. This proof is much shorter than the original one and bypasses one of the main difficulties, which may be called the extension problem, specific to Torelli groups as opposed to the Teichm\"uller modular groups. The extension problem is discussed in details in the second paper of this series.