On the Lie algebras of surface pure braid groups

TL;DR

This paper investigates the Lie algebra structure of pure braid groups on closed surfaces, extending previous rational results to an integral setting, thereby deepening understanding of their algebraic properties.

Contribution

The authors prove that Bezrukavnikov's rational presentation of the Lie algebra associated with surface pure braid groups also holds integrally, not just over the rationals.

Findings

The integral version of Bezrukavnikov's presentation is valid.

The Lie algebra structure is fully described over integers.

This extends the understanding of surface pure braid groups' algebraic properties.

Abstract

We consider the Lie algebra associated with the descending central series filtration of the pure braid group of a closed surface of arbitrary genus. R. Bezrukavnikov gave a presentation of this Lie algebra over the rational numbers. We show that his presentation remains true for this Lie algebra itself, i.e. over integers.