On presentations of surface braid groups

Abstract

We give presentations of braid groups and pure braid groups on surfaces and we show some properties of surface pure braid groups.

Bi-invariant metrics and quasi-morphisms on groups of Hamiltonian diffeomorphisms of surfaces

“…it twists only i-th and j-th strands. Each generator in the set S \ S ′ naturally corresponds to one of the generators of π 1 (Σ g , z i ) described above, see [4].…”

“…Denote this set by QCal g . It is infinite and it does not contain non-trivial homomorphisms, since every homomorphism on B 2 (Σ g ) must be trivial on B 2 , see [4].…”

A family of representations of braid groups on surfaces

“…The study of braid groups on surfaces has been revived recently. González-Meneses [2001] found new presentations of the braid groups on surfaces, and the authors of [Bellingeri 2004;Bellingeri and Godelle 2007] found positive presentations of the braid groups B n,k ( ) for all surfaces , with or without boundary. Here, we are interested in braid groups on surfaces with nonempty boundary and will use Bellingeri's presentations.…”

“…Then there are continuous maps i : q → and j : → q that are homotopy inverses each other. The induced mapsī : B n+q,k ( ) → B n,k ( ) andj : B n,k ( ) → B n+q,k ( ) on configuration spaces are also homotopy inverses each other and induce isomorphismsī * andj * on fundamental groups [Bellingeri 2004;Paris and Rolfsen 1999]. Therefore we may assume = (g, 1) by treating all but one boundary component as a puncture whenever we deal with a surface with nonempty boundary.…”

On the structure of braid groups on complexes

“…Indeed, for a given surface Σ, one can extract geometric information from its braid group as follows. The proof is obvious by the group presentation for B n (Σ), see [5], and we omit the proof.…”