Orderability theory for braid groups over surfaces and for link-homotopy generalized string-links over surfaces

• Grant number: 14/27135-6
• Support Opportunities: Scholarships abroad - Research Internship - Post-doctor
• Start date: June 10, 2015
• End date: June 09, 2016
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Denise de Mattos
• Grantee: Juliana Roberta Theodoro de Lima
• Supervisor: Steven Boyer
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Institution abroad: Université du Québec à Montréal (UQÀM), Canada
• Associated to the scholarship: 14/08088-7 - Orderability theory for braid groups over surfaces and for Link-Homotopy generalized String-Links over surfaces. Representation theorem for the Link-Homotopy generalized String-Links over surfaces, BP.PD

Abstract

An important area that has been developed is called "braid groups on surfaces with genus": the presentation theorem of Gonzalez-Meneses for the braid groups on surfaces is a generalization of Artin's presentation for the braid groups over the disk. In his recent work, Gonzalez-Meneses found the smallest presentation for braid groups over surfaces. In particular, he produces a presentation for the braid groups over orientable surfaces of genus g greater or equal than 1 and over non orientable surfaces of genus g greater or equal than 2. In particular, this paper was the central paper of study in the master's degree of the mentioned student. Recently the representation theorem for the braid groups over the disk was also generalized to braid groups over surfaces by Bardakov and Belingeri. Recently Rolfsen, Dynnikov, Dehornoy and Wiest, demonstrated topological reasons for the existence of a left-ordering of the braid groups over the disk, i.e, there is a strict total ordering of the braids that is invariant under multiplication from the left. They also showed the pure braid groups over the unit disk are bi-orderable, i.e., there is a left and right invariant strict total ordering for this group. Later, Gonzalez-Meneses proved that the pure braid groups over surfaces are bi-orderable too. Along these same lines, we can study generalized string-links, which informally are generalizations of braids. The difference is that for the former, we consider embedded strands up to link-homotopy, whereas for the latter, we consider embedded strands up to isotopy. We say that generalized string links are generalizations of braids since a link homotopy always involves a finite numbers of isotopies and crossing changes. In the PhD thesis of the applicant, she proved that the homotopy generalized string links over orientable surfaces of genus g greater or equal than 1 form a group, and calculated a presentation for the group. Her thesis also generalizes the results of Yurasovskaya, by showing that the homotopy string-links over orientable surfaces of genus g greater or equal than 1 form a bi-orderable group. Her thesis was advised by Denise de Mattos and co-advised by Rolfsen at University of São Paulo in Brazil and at University of British Columbia in Canada, respectively. Her work represents a significant step towards determining the orderability properties of these groups, and has been submitted for publication. In this research proposal, we will outline how to develop generalizations for the results mentioned above. More specifically, we have as aim the resolution of the following problems related to this theme: 1. Prove that the braid group on surfaces is (or is not) left-orderable. 2. Prove that the link-homotopy group of generalized string link on surfaces is (or is not) left-orderable. (AU)