Periodic Reporting for period 1 - Rabgres (Ramification of braid groups and their related structures)
Periodo di rendicontazione: 2022-10-01 al 2024-09-30
In the RABGRES project, the researcher Neha Nanda undertook the fellowship at the Laboratoire de Mathématiques Nicolas Oresme (LMNO), Université de Caen Normandie, under the supervision of the host Professor John Guaschi, focusing on the ramifications of braid groups and their related structures. Braid groups, initially introduced by Emil Artin, have become foundational objects in several branches of mathematics and theoretical physics, including low-dimensional topology, algebraic geometry, dynamical systems, and quantum physics. They are deeply intertwined with knot theory, which studies the classification of knots and has led to the development of important knot invariants such as the Alexander and Jones polynomials. These braid groups, and their generalisations, offer vast avenues of exploration due to their structural richness and their applications in both pure and applied mathematics.
One of the project's key directions was the study of Khovanov's twin groups, which serve as a planar analogue of braid groups, focusing on configurations of strands in a plane. These twin groups are essential for understanding doodles—classes of Jordan curves on the sphere that do not have triple intersections—as explored by Fenn and Taylor. A few related groups which are investigated are cactus groups (twin groups embed in the cactus groups), virtual twin groups (related to doodles on closed oriented surfaces), right-angled Coxeter groups (general class of groups to which twin groups belong), right-angled Artin groups (obtained from the presentation of right-angled Coxeter groups), diagram groups (pure twin groups can be viewed as diagram groups).
Even though classical braid groups and some of their generalisations have apparently similar presentations and topological interpretations, it is important to be able to distinguish them and understand how they differ algebraically, and it is interesting to explore the above-mentioned groups independently.
The main objectives of the project are as follows.
1. Understand better the (pure) twin groups by studying some of their invariants, such as their virtual cohomological dimension, and look for good presentations for them using Fadell-Neuwirth-like short exact sequences.
2. Following the construction of surface braid groups from Artin braid groups, generalise the (pure) twin groups to surface (pure) twin groups, obtain presentations of these groups, and seek doodle-like structures to which they correspond.
3. Construct new (virtual) doodle invariants, some being analogous to those known for knots and links, the aim is to classify (virtual) doodles.
The project aimed to uncover deeper algebraic and topological properties of these structures, contributing to a dynamic and rapidly evolving research area.
One of the project's key directions was the study of Khovanov's twin groups, which serve as a planar analogue of braid groups, focusing on configurations of strands in a plane. These twin groups are essential for understanding doodles—classes of Jordan curves on the sphere that do not have triple intersections—as explored by Fenn and Taylor. A few related groups which are investigated are cactus groups (twin groups embed in the cactus groups), virtual twin groups (related to doodles on closed oriented surfaces), right-angled Coxeter groups (general class of groups to which twin groups belong), right-angled Artin groups (obtained from the presentation of right-angled Coxeter groups), diagram groups (pure twin groups can be viewed as diagram groups).
Even though classical braid groups and some of their generalisations have apparently similar presentations and topological interpretations, it is important to be able to distinguish them and understand how they differ algebraically, and it is interesting to explore the above-mentioned groups independently.
The main objectives of the project are as follows.
1. Understand better the (pure) twin groups by studying some of their invariants, such as their virtual cohomological dimension, and look for good presentations for them using Fadell-Neuwirth-like short exact sequences.
2. Following the construction of surface braid groups from Artin braid groups, generalise the (pure) twin groups to surface (pure) twin groups, obtain presentations of these groups, and seek doodle-like structures to which they correspond.
3. Construct new (virtual) doodle invariants, some being analogous to those known for knots and links, the aim is to classify (virtual) doodles.
The project aimed to uncover deeper algebraic and topological properties of these structures, contributing to a dynamic and rapidly evolving research area.
The objective of the project focussed on twin groups, geometric analogues of braids on the plane, and their various generalisations and related objects. The aim is to investigate the algebraic and geometric structures of these groups, their classification, and the study of related invariants. A brief description towards meeting the objectives is provided below.
1. (Right-Angled Artin Groups (RAAGs) and Symmetric Diagram Groups) The twin group, or planar braid group, is connected to diagram groups, and its pure twin subgroup may be viewed as a diagram group. The researcher established that the pure virtual twin group is a RAAG and also a symmetric diagram group. This motivated the proof of the following general result: every finitely generated RAAG is a symmetric diagram group. The work is published online on arXiv and has been submitted to a peer-reviewed journal. This is joint work with Paolo Bellingeri (Université de Caen Normandie) and Anthony Genevois (Université de Montpellier).
2. (Algebraic Properties of Cactus Groups) The researcher investigated the algebraic and combinatorial properties of cactus groups (which contain twin groups), specifically their quotients and lower central series. It is proved that the cactus groups share similar structural properties with right-angled Coxeter groups (RACGs) of which twin groups are examples. Through this work, she provided new insights into the behaviour of these groups and their lower central series, contributing to a deeper understanding of their internal algebraic structure. The work is published online on arXiv and has been submitted to a peer-reviewed journal. This is a joint work with Hugo Chemin (Université de Caen Normandie).
3. (Commutator Subgroups and Crystallographic Quotients) Presentations of braid and twin groups may be obtained by omitting certain relations from the standard presentation of symmetric groups. The picture is completed through triplet groups. In order to have a better understanding of the structure of virtual twin (triplet) groups, the researcher investigated their commutator subgroups (obtained explicit presentations) and crystallographic quotients. The research article has been published in the Journal of Pure and Applied Algebra (2024). This is a joint work with Mahender Singh, Pravin Kumar (IISER Mohali) and Tushar Naik (NISER Bhubaneswar).
4. (Lower Central Series of RACGs) The lower central series of the RACGs have been investigated by the researcher and the host professor. In particular, progress has been made till the fourth consecutive quotients of lower central series of RACGs , and their explicit ranks have been computed. The work is in the draft preparation stage and is in collaboration with John Guaschi (the host).
5. (Quasitoric Braid Presentations of Generalised Knots) The researcher defined the notion of a quasitoric normal generalised braid and proved that every oriented normal generalised knot is the closure of a quasitoric normal generalised braid. This work is in the draft preparation stage and is in collaboration with Manpreet Singh (University of South Florida).
6. (Non-Abelian Quotients of Virtual Artin Groups) Virtual Artin groups have recently been defined as a generalisation of virtual braid groups, by replacing the symmetric group with a Coxeter group. One of the motivations for studying finite homomorphisms and finite quotients is to distinguish finitely presented groups via their quotients. One of the main results of the work is the computation of the smallest non-abelian quotient for the virtual Artin group corresponding to Coxeter groups of finite type. These quotients for virtual Artin groups relate to the smallest quotient of the underlying Coxeter groups. This is a joint work with John Guaschi (the host) and Oscar Ocampo (Universidade Federal de Bahia).
1. (Right-Angled Artin Groups (RAAGs) and Symmetric Diagram Groups) The twin group, or planar braid group, is connected to diagram groups, and its pure twin subgroup may be viewed as a diagram group. The researcher established that the pure virtual twin group is a RAAG and also a symmetric diagram group. This motivated the proof of the following general result: every finitely generated RAAG is a symmetric diagram group. The work is published online on arXiv and has been submitted to a peer-reviewed journal. This is joint work with Paolo Bellingeri (Université de Caen Normandie) and Anthony Genevois (Université de Montpellier).
2. (Algebraic Properties of Cactus Groups) The researcher investigated the algebraic and combinatorial properties of cactus groups (which contain twin groups), specifically their quotients and lower central series. It is proved that the cactus groups share similar structural properties with right-angled Coxeter groups (RACGs) of which twin groups are examples. Through this work, she provided new insights into the behaviour of these groups and their lower central series, contributing to a deeper understanding of their internal algebraic structure. The work is published online on arXiv and has been submitted to a peer-reviewed journal. This is a joint work with Hugo Chemin (Université de Caen Normandie).
3. (Commutator Subgroups and Crystallographic Quotients) Presentations of braid and twin groups may be obtained by omitting certain relations from the standard presentation of symmetric groups. The picture is completed through triplet groups. In order to have a better understanding of the structure of virtual twin (triplet) groups, the researcher investigated their commutator subgroups (obtained explicit presentations) and crystallographic quotients. The research article has been published in the Journal of Pure and Applied Algebra (2024). This is a joint work with Mahender Singh, Pravin Kumar (IISER Mohali) and Tushar Naik (NISER Bhubaneswar).
4. (Lower Central Series of RACGs) The lower central series of the RACGs have been investigated by the researcher and the host professor. In particular, progress has been made till the fourth consecutive quotients of lower central series of RACGs , and their explicit ranks have been computed. The work is in the draft preparation stage and is in collaboration with John Guaschi (the host).
5. (Quasitoric Braid Presentations of Generalised Knots) The researcher defined the notion of a quasitoric normal generalised braid and proved that every oriented normal generalised knot is the closure of a quasitoric normal generalised braid. This work is in the draft preparation stage and is in collaboration with Manpreet Singh (University of South Florida).
6. (Non-Abelian Quotients of Virtual Artin Groups) Virtual Artin groups have recently been defined as a generalisation of virtual braid groups, by replacing the symmetric group with a Coxeter group. One of the motivations for studying finite homomorphisms and finite quotients is to distinguish finitely presented groups via their quotients. One of the main results of the work is the computation of the smallest non-abelian quotient for the virtual Artin group corresponding to Coxeter groups of finite type. These quotients for virtual Artin groups relate to the smallest quotient of the underlying Coxeter groups. This is a joint work with John Guaschi (the host) and Oscar Ocampo (Universidade Federal de Bahia).
The research has advanced the understanding of algebraic structures related to braid groups such as diagram groups, cactus groups, and virtual Artin groups, contributing to the state of the art in these fields. Since the project lies in the domain of pure mathematics, major potential impacts include the introduction of new objects to investigate, building up the theory by providing the foundational results, and introducing and formalising techniques to deal with problem-solving. While pure mathematics doesn’t necessarily start with an immediate application to the physical world, over time, abstract theories often find profound applications over a period of time. These lead to future research projects and collaborations for the advancement of the subject. The following summarises the key scientific achievements and innovation outputs:
1. (Right-Angled Artin Groups and Symmetric Diagram Groups) This work gives a new perspective to RAAGs and on the other hand, proves that a big class of groups are examples of the symmetric diagram groups.
2. (Algebraic Properties of Cactus Groups) This work advances the knowledge of group presentations and quotients, particularly in relation to Coxeter groups.
3. (Commutator Subgroups and Crystallographic Quotients) Advances the understanding of virtual extensions of symmetric groups and their algebraic properties.
4. (Lower Central Series of RACGs) This study relates graph-theoretic properties to group theory, deepening the understanding of Coxeter groups and their central series, the lower central series being used as a group theoretic invariant to further distinguish the groups.
5. (Quasitoric Braid Presentations of Generalised Knots) Extends the notion of quasitoric braids in knot theory to generalised knots, and introduces generalised braid theory. It paves the way to investigate all braid theories simultaneously and to further compute explicit presentations of the quasitoric subgroups of generalised braid groups.
6. (Non-Abelian Quotients of Virtual Artin Groups) This work provides new insights into the quotient structures of virtual Artin groups, with potential implications for group classification.
1. (Right-Angled Artin Groups and Symmetric Diagram Groups) This work gives a new perspective to RAAGs and on the other hand, proves that a big class of groups are examples of the symmetric diagram groups.
2. (Algebraic Properties of Cactus Groups) This work advances the knowledge of group presentations and quotients, particularly in relation to Coxeter groups.
3. (Commutator Subgroups and Crystallographic Quotients) Advances the understanding of virtual extensions of symmetric groups and their algebraic properties.
4. (Lower Central Series of RACGs) This study relates graph-theoretic properties to group theory, deepening the understanding of Coxeter groups and their central series, the lower central series being used as a group theoretic invariant to further distinguish the groups.
5. (Quasitoric Braid Presentations of Generalised Knots) Extends the notion of quasitoric braids in knot theory to generalised knots, and introduces generalised braid theory. It paves the way to investigate all braid theories simultaneously and to further compute explicit presentations of the quasitoric subgroups of generalised braid groups.
6. (Non-Abelian Quotients of Virtual Artin Groups) This work provides new insights into the quotient structures of virtual Artin groups, with potential implications for group classification.