Simplicial structures and normal forms for mapping class groups and braid groups

Abstract

In this paper we show that the mapping class groups of any surface with nonempty boundary form a simplicial group as the number of marked points varies. This extends the simplicial structure on braid groups of surfaces found by Berrick, Cohen, Wong and Wu. We use the simplicial maps to construct compatible normal forms for elements of the braid groups and mapping class groups of surfaces with boundary.

Brunnian subgroups of mapping class groups and braid groups

“…Apart from the proof of Proposition 3.7, we do not consider the full mapping class groups here. Again, there is another extension of this result in [3], where it is shown that when the surface M has nonempty boundary then the pure (respectively, full) mapping class groups admit degeneracy maps as well as face maps, and therefore form a simplicial group (respectively, crossed simplicial group).…”

“…We do not consider full braid groups in this paper. Another extension of this result occurs in [3], where it is shown that, when the surface M admits a nonvanishing vector field, the pure (respectively, full) braid groups may be endowed with degeneracy maps as well as face maps, and therefore form a simplicial group (respectively, crossed simplicial group). Next, we recall the Δ-group structure on the mapping class groups of a surface M .…”