The virtual cohomology dimension of Teichm\"uller modular groups: the first results and a road not taken

TL;DR

This paper provides a detailed exposition and full proofs of earlier results on the virtual cohomology dimension of Teichmüller modular groups, emphasizing the use of the Hatcher-Thurston complex and discussing future directions.

Contribution

It offers a comprehensive presentation of 1983 results, including proofs and context, highlighting a novel approach using the simply-connectedness of the Hatcher-Thurston complex.

Findings

Results on the virtual cohomology dimension of Teichmüller groups

Use of the Hatcher-Thurston complex's simply-connectedness

Discussion of potential future research directions

Abstract

The paper is devoted to a detailed exposition of results outlined in author's 1983 note "On the virtual cohomology dimension of the Teichm\"uller modular group". The paper includes the full proofs and a discussion of the context which motivated both the results and the methods used. The reminiscences included into the last section are an integral part of this discussion. A key idea is to use the simply-connectedness of the Hatcher-Thurston complex. This idea was hardly developed further, but still keeps a promise.