Research Projects

The research project focuses on groups in low-dimensional topology, combinatorial group theory, finite group theory, and (nilpotent) Lie algebra. It consists of two parts: exploring pure twin groups, which arise naturally in low-dimensional topology, and the isomorphism problem in Coxeter groups. Braid groups, which are connected to knots, are celebrated objects in various fields such as mathematics, theoretical physics, chemistry, and biology. Twin groups, planar and analogous to braid groups, play a crucial role in studying doodles, the isotopy classes of immersed circles on surfaces. Recently, the study of doodles has been extended to immersed circles on closed-oriented surfaces, forming a new family of groups called virtual twin groups. The project aims to expand expertise in pure twin groups, which are among the most important normal subgroups or twin groups. Pure twin groups can also be realized as fundamental groups of the complement of triple diagonals in Euclidean spaces. The isomorphism problem in Coxeter groups is a fundamental problem in group theory, as it asks whether two presentations of groups can be isomorphic. Coxeter groups arise naturally in the study of reflections of Euclidean spaces and find applications in various areas of mathematics and theoretical physics. To generalize the isomorphism problem in Coxeter groups, the researchers plan to study the automorphism groups of a family of (odd) Coxeter groups whose associated Coxeter graphs are trees.