Subgroups in Artin Groups and Lattices in Products of Trees

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A group is an algebraic structure encoding symmetries of an object. It can be defined abstractly, as a collection of strings of letters, where certain equations describe which two strings correspond to the same symmetry. Such letters are called generators, and the equations are called relations, and together they form what is called a group presentation. Geometric group theory studies the connection between the geometry of the object, and the properties of the group of its symmetries. An example of a group is the set of integers, which can be viewed as symmetries of a line, where a positive number moves points on the line to the right, and a negative number to the left. A subgroup of a group is a smaller collection of symmetries, closed under composition. In the group of integers, an example of a subgroup is the collection of the symmetries moving by an even distance. Understanding the subgroup structure is essential in studying the whole group. This project will address questions about subgroups with prescribed properties in two families of groups: Artin groups and lattices in products of trees. Groups in both of those families can be described by simple looking presentations, but many questions about them remain unanswered. The project will also promote the participation of women in mathematics via mentoring and outreach.The first goal of this project is to examine the actions of Artin groups on CAT(0) cube complexes. This project will investigate for which Artin groups is every group element is separated by some codimension-1 subgroup, and for which of them this leads to proper actions on CAT(0) cube complexes. The theory of CAT(0) cube complexes, and special cube complexes in particular, has been a fruitful tool in understanding groups. Proving that Artin groups act properly on CAT(0) cube complexes would answer many outstanding questions about Artin groups; for example, it could provide a solution to the word problem. The PI will also continue her work on the residual finiteness of Artin groups in this project. In the second project, the PI will study cocompact lattices in products of trees and their subgroup structures. In particular, the PI will determine if all such groups are incoherent. Showing that all lattices in a product of trees are incoherent would be an indication that coherence is a quasi-isometry invariant. The project will also determine if any two infinite order elements in a lattice in a product of trees either commute or generate a free subgroup, when raised to high powers. The project also includes training and mentoring of undergraduate and graduate students with an emphasis on broadening participation of women in mathematics. The PI is also planning a collaborative educational project with Jankiewicz Studio, a design firm specializing in educational and cultural projects at the intersection of design, art, science and technology.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.