Research

In mathematics and abstract algebra, group theory studies the algebraic structure known as groups. A group is a set equipped with a binary operation that combines any two elements to form a third in such a way that three axioms hold namely associativity, identity and invertibility. Groups with a finite number of elements can be described by writing down the group table, that is, a table consisting of all possible combinations of elements of the group. However, such techniques do not apply to groups with an infinite number of elements. A more compact way of defining a group is something called group presentation. A group presentation is a description of a group in terms of a set of generators and a set of relations amongst them. Combinatorial group theory is a branch of group theory that studies groups from their presentations. The area makes use of the connection of graphs associated with the presentation such as Cayley graphs and star graphs, to understand the structure of the group.

A key problem in this subject area is to describe the properties of a group from a given presentation that defines it. The word problem asks whether two words in the generators of a group are effectively the same group element. A more difficult problem known as the isomorphism problem asks whether two different presentations define isomorphic groups. Research has shown that in the most general setting, most of the questions that mathematicians might want to ask are unsolvable. For this reason, it is necessary to focus on particular classes of presentations. There are two major classes of presentations that feature in my research namely cyclic presentations and one-relator product presentations, and these define cyclically presented groups and one-relator product of groups respectively.

Articles

1. Chinyere, Ihechukwu and Williams, Gerald. Hyperbolicity of T(6) cyclically presented groups, Groups Geom. Dyn., 16: 341--361, 2022 ems.press/journals/ggd/articles/4552494

2. Chinyere, Ihechukwu and Williams, Gerald. Generalized polygons and star graphs of cyclic presentations of groups, J. Combin. Theory Ser. A, 190: Paper No. 105638, 2022 doi.org/10.1016/j.jcta.2022.105638

3. Chinyere, Ihechukwu and Williams, Gerald. Hyperbolic groups of Fibonacci type and T(5) cyclically presented groups, J. Algebra, 580:104--126, 2021 doi.org/10.1016/j.jalgebra.2021.04.003

4. Chinyere, Ihechukwu and Williams, Gerald. Fractional Fibonacci groups with an odd number of generators, Topology Appl., 312: Paper No. 108083, 16, 2022 doi.org/10.1016/j.topol.2022.108083

5. Chinyere, Ihechukwu and Bainson, Bernard Oduoku. Perfect Prishchepov groups. J. Algebra, 588: 515--532, 2021 doi.org/10.1016/j.jalgebra.2021.09.005

6. Chinyere, Ihechukwu. Structure of words with short 2-length in a free product of groups, J. Algebra, 519: 312--324, 2019 doi.org/10.1016/j.jalgebra.2018.11.005

7. Chinyere, Ihechukwu and Howie, James. On one-relator products induced by generalized triangle groups II, Comm. Algebra, 46: 1464--1475, 2018 doi.org/10.1080/00927872.2017.1346772

8. Chinyere, Ihechukwu and Howie, James. On one-relator products induced by generalized triangle groups I, Comm. Algebra, 46: 1138--1154, 2018 doi.org/10.1080/00927872.2017.1339057

9. Chinyere, Ihechukwu and Howie, James. Non-triviality of some one-relator products of three groups, Internat. J. Algebra Comput., 26: 533--550, 2016 doi.org/10.1142/S0218196716500223

10. Chinyere, Ihechukwu and Williams, Gerald. Redundant relators in cyclic presentations of groups, 2021 arxiv.org/abs/2112.10538

I found these videos of me talking about the paper 'Hyperbolic groups of Fibonacci type and T(5) cyclically presented'.