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The paper is devoted to a detailed exposition of results outlined in author's1983 note "On the virtual cohomology dimension of the Teichm\"uller modulargroup". The paper includes the full proofs and a discussion of the contextwhich motivated both the results and the methods used. The reminiscencesincluded into the last section are an integral part of this discussion. A keyidea is to use the simply-connectedness of the Hatcher-Thurston complex. Thisidea was hardly developed further, but still keeps a promise.

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