Asymptotic and cohomological dimension of surface braid groups and poly-surface groups

TL;DR

This paper computes the asymptotic dimension of all surface braid groups and poly-surface groups, showing they coincide with their virtual cohomological dimension, and establishes their duality properties.

Contribution

It extends the understanding of asymptotic and cohomological dimensions to non-orientable, infinite-type, and poly-surface groups, revealing their duality structures.

Findings

Asymptotic dimension equals virtual cohomological dimension for all surface braid groups.

Surface braid groups of finite-type surfaces are virtual duality groups.

Infinite-type surface braid groups are countable and normally poly-free.

Abstract

In this paper, we determine the asymptotic dimension for all surface braid groups -- including those associated with non-orientable and infinite-type surfaces -- as well as for torsion-free poly-finitely generated surface groups. We demonstrate that for both classes, the virtual cohomological dimension and the asymptotic dimension coincide. For poly-finitely generated surface groups and braid groups of finite-type surfaces, our approach establishes that these groups are virtual duality groups in the sense of Bieri-Eckmann. In the case of infinite-type surfaces, the argument is based on the fact that their braid groups are countable and normally poly-free.