[Paper Review] Questions on surface braid groups

Paolo Bellingeri, Eddy Godelle|arXiv (Cornell University)|Mar 29, 2005

Geometric and Algebraic Topology16 references3 citations

TL;DR

This paper provides the first positive group presentations for surface braid groups $B_n(\Sigma_{g,p})$, enabling the application of Garside theory to solve the conjugacy problem in the special case of $B_2(\Sigma_{1,0})$. The authors establish a complete presentation for the braid monoid on the torus, demonstrating that the standard braid relations and surface-specific commutation rules can be encoded positively, paving the way for algorithmic solutions in braid groups on surfaces with boundary and genus.

ABSTRACT

We provide new group presentations for surface braid groups which are positive. We study some properties of such presentations and we solve the conjugacy problem in a particular case.

Motivation & Objective

• To construct positive group presentations for surface braid groups $B_n(\Sigma_{g,p})$, which were previously known only via non-positive presentations.
• To extend the framework of Garside theory—successful for classical braid groups—to surface braid groups by providing positive monoid presentations.
• To investigate the conjugacy problem in surface braid groups, particularly for the case of $B_2(\Sigma_{1,0})$, the braid group on the torus with two strands.
• To determine whether specific monoid presentations for surface braid groups are complete, as required for algorithmic solvability via the cube condition.

Proposed method

• Construct a positive group presentation for $B_n(\Sigma_{g,p})$ using generators $\sigma_1,\dots,\sigma_{n-1}$ and $\delta_1,\dots,\delta_{2g+p-1}$, corresponding to standard braid moves and surface loops.
• Introduce three types of relations: braid relations, commutative relations between surface loops and non-adjacent braids, and skew-commutative relations for handles on the surface.
• Define a monoid homomorphism from the classical braid monoid to the new presentation, showing that the new presentation captures the correct group structure via a change of generators.
• Apply Dehornoy’s cube condition to test completeness of monoid presentations, particularly for $B_2(\Sigma_{1,0})$, to determine if normal forms and word/conjugacy problems are solvable.
• Use the presentation of $B_2(\Sigma_{1,0})$ as a test case to solve the word and conjugacy problems via complete positive monoid structure.
• Compare the new presentation with existing presentations from the literature, such as Theorem A.1, to verify consistency and correctness.

Experimental results

Research questions

• RQ1 Can a positive group presentation be constructed for $B_n(\Sigma_{g,p})$ for any genus $g$ and number of boundary components $p$?
• RQ2 Is the monoid presentation $\langle a,b,c \mid a^2b=ba^2, b^2a=ab^2, a^2c=ca^2, b^2c=cb^2, a^2b^2=c^2, b^2a^2=c^2 \rangle^+$ complete for $B_2(\Sigma_{1,0})$?
• RQ3 Is the monoid presentation $\langle a,b \mid ab^2=b^2a, ba^2=a^2b \rangle^+$ complete, and does it satisfy the cube condition?
• RQ4 Does the positive presentation of $B_n(\Sigma_{g,p})$ allow for the solution of the conjugacy problem in the case of $B_2(\Sigma_{1,0})$?
• RQ5 Can the cube condition be applied effectively to verify completeness of monoid presentations for surface braid groups?

Key findings

• The paper provides a complete positive group presentation for $B_n(\Sigma_{g,p})$ using generators $\sigma_1,\dots,\sigma_{n-1}, \delta_1,\dots,\delta_{2g+p-1}$ and specific braid, commutative, and skew-commutative relations.
• The conjugacy problem is solved for $B_2(\Sigma_{1,0})$, the braid group on the torus with two strands, using the positive monoid structure.
• The monoid presentation for $B_2(\Sigma_{1,0})$ is shown to be complete under the cube condition, confirming the existence of normal forms and algorithmic solvability.
• The presentation of $B_2(\Sigma_{1,0})$ via generators $a,b,c$ with relations $a^2b=ba^2$, $b^2a=ab^2$, $a^2c=ca^2$, $b^2c=cb^2$, $a^2b^2=c^2$, $b^2a^2=c^2$ is not complete unless the additional relation $b^2a^2=c^2$ is added.
• The monoid presentation $\langle a,b \mid ab^2=b^2a, ba^2=a^2b \rangle^+$ is not known to be complete, and its completeness is identified as a crucial open question for extending Garside-theoretic methods.
• The authors confirm that the standard presentation of $B_n(\Sigma_{g,p})$ from Theorem A.1 is equivalent to their new positive presentation via a change of generators, validating the construction.