Periodic Reporting for period 1 - MapSurf (Combinatorial and Geometric Methods for Mapping Class Groups of Surfaces)
Berichtszeitraum: 2023-09-01 bis 2025-08-31
In this project, we study groups called braid groups. Suppose we have a disc, and we fix n points in that disc, where n is some whole number. Let’s call this disc with n marked points D_n. We consider symmetries of this disc D_n. In this context (geometric topology) a symmetry is a transformation called a homeomorphism taking D_n back to itself. We require that the homeomorphism takes each of the n marked points to a marked point, but not necessarily to the same one. The braid group on n strands is the group of all homeomorphisms from D_n to itself, up to a natural equivalence.
To visualize an example of a homeomorphism of D_n, imagine that at the left of a round cake tin, we pour vanilla batter, and at the right of the tin we pour chocolate batter. At n points in the tin, we insert a vertical skewer, and we start dragging the skewers through the cake mix, always bringing the collection of n skewers back to the same n points. Intuitively, we understand that the vanilla and chocolate batters will start to swirl together, and the more we repeat our motions, the more the two batters will get mixed. This is the principle we use to study homeomorphisms of D_n: we have some subset that is fixed at the beginning and we look at how it gets “mixed” by our homeomorphisms. However, there is an important difference with our cake example: in practice we can’t separate two cake batters which have been mixed together. On the contrary, with homeomorphisms we can also “unmix”. Indeed, every homeomorphism has an inverse which undoes its effects.
Braid groups are a special case of mapping class groups of surfaces. The disc D_n with n marked points is special among surfaces because we can draw it in the plane. However, a generic surface cannot be drawn in the plane. For example, a torus is a surface that looks like a rubber ring or the surface of a doughnut. We have surfaces that look like doughnuts with arbitrarily many “holes”, and we call the number of “holes” the genus of the surface. Surfaces of genus at least 1 cannot be drawn in the plane. For surfaces of positive genus, we can still consider the group of symmetries of the surfaces, which we call the mapping class group of the surface. The fact that the disc D_n with n marked points can be drawn in the plane means that we can study braid groups in a more hands-on way than other mapping class groups.
A turning point in the study of mapping class groups of surfaces was a paper of Masur and Minsky in 2000, which fits into the setting of geometric group theory. This field of mathematics aims to understand groups (coming from algebra), by how they act on spaces (coming from geometry). Masur and Minsky studied the mapping class group of a surface by looking at its action on certain geometric spaces associated to the surface. A key idea in their work is that for each surface we have a hierarchy of spaces associated to the surface, and we can understand homeomorphisms by how they act on the different spaces in this hierarchy.
Masur and Minsky’s hierarchy machinery is integral in the modern study of mapping class groups, and has inspired major new directions in geometric group theory (for example, the theory of hierarchically hyperbolic spaces). Although the hierarchy machinery has been the topic of much research, it can be quite daunting to researchers who come to it for the first time. The topic of our project is to leverage the hands-on nature of the braid group to give a simple and visual description of how a hierarchy might play out in this special case.
To visualize an example of a homeomorphism of D_n, imagine that at the left of a round cake tin, we pour vanilla batter, and at the right of the tin we pour chocolate batter. At n points in the tin, we insert a vertical skewer, and we start dragging the skewers through the cake mix, always bringing the collection of n skewers back to the same n points. Intuitively, we understand that the vanilla and chocolate batters will start to swirl together, and the more we repeat our motions, the more the two batters will get mixed. This is the principle we use to study homeomorphisms of D_n: we have some subset that is fixed at the beginning and we look at how it gets “mixed” by our homeomorphisms. However, there is an important difference with our cake example: in practice we can’t separate two cake batters which have been mixed together. On the contrary, with homeomorphisms we can also “unmix”. Indeed, every homeomorphism has an inverse which undoes its effects.
Braid groups are a special case of mapping class groups of surfaces. The disc D_n with n marked points is special among surfaces because we can draw it in the plane. However, a generic surface cannot be drawn in the plane. For example, a torus is a surface that looks like a rubber ring or the surface of a doughnut. We have surfaces that look like doughnuts with arbitrarily many “holes”, and we call the number of “holes” the genus of the surface. Surfaces of genus at least 1 cannot be drawn in the plane. For surfaces of positive genus, we can still consider the group of symmetries of the surfaces, which we call the mapping class group of the surface. The fact that the disc D_n with n marked points can be drawn in the plane means that we can study braid groups in a more hands-on way than other mapping class groups.
A turning point in the study of mapping class groups of surfaces was a paper of Masur and Minsky in 2000, which fits into the setting of geometric group theory. This field of mathematics aims to understand groups (coming from algebra), by how they act on spaces (coming from geometry). Masur and Minsky studied the mapping class group of a surface by looking at its action on certain geometric spaces associated to the surface. A key idea in their work is that for each surface we have a hierarchy of spaces associated to the surface, and we can understand homeomorphisms by how they act on the different spaces in this hierarchy.
Masur and Minsky’s hierarchy machinery is integral in the modern study of mapping class groups, and has inspired major new directions in geometric group theory (for example, the theory of hierarchically hyperbolic spaces). Although the hierarchy machinery has been the topic of much research, it can be quite daunting to researchers who come to it for the first time. The topic of our project is to leverage the hands-on nature of the braid group to give a simple and visual description of how a hierarchy might play out in this special case.
We consider the braid group B_n on n strands, that is, the mapping class group of the n-times punctured disc D_n. We study B_n using curve diagrams on D_n. We make this explicit by fixing an embedding of D_n into the Euclidean plane such that the boundary of D_n is a round circle with a diameter on the real line, and such that the n marked points lie on the real line. We fix a trivial curve diagram which is defined by vertical arcs between consecutive punctures (our arcs are defined up to homotopy, roughly meaning that if one arc can be continuously deformed into another then we consider them to be the same arc). If we apply a braid (homeomorphism) to D_n, then our curve diagram is changes. If the braid we apply is non-trivial, then the new curve diagram will appear “more tangled”, in the sense that it will have more intersections with the real line than the trivial curve diagram has. Starting from the tangled curve diagram, we apply an algorithm to untangle the curve diagram step by step, until we are back at the trivial curve diagram. The algorithm we use is inspired by work of Dynnikov and Wiest, and was previously studied by Caruso in a more specific context.
In order to prove results about the algorithm, we apply a combination of combinatorial and geometric methods. We use a geometric object called the curve graph of a surface, which is ubiquitous in the study of the large-scale geometry of the mapping class group. We study the active subsurfaces associated to the braid we have applied. These are those subsurfaces where the distance in the curve graph between the tangled curve diagram and the trivial curve diagram is large. These subsurfaces are important in Masur and Minsky’s hierarchy machinery. We prove that we can actually see these subsurfaces in our picture as the algorithm progresses: at some point the subsurface becomes “almost round” in a precise sense, and it stays almost round until the curve diagram restricted to the subsurface has been untangled.
In order to prove results about the algorithm, we apply a combination of combinatorial and geometric methods. We use a geometric object called the curve graph of a surface, which is ubiquitous in the study of the large-scale geometry of the mapping class group. We study the active subsurfaces associated to the braid we have applied. These are those subsurfaces where the distance in the curve graph between the tangled curve diagram and the trivial curve diagram is large. These subsurfaces are important in Masur and Minsky’s hierarchy machinery. We prove that we can actually see these subsurfaces in our picture as the algorithm progresses: at some point the subsurface becomes “almost round” in a precise sense, and it stays almost round until the curve diagram restricted to the subsurface has been untangled.
We study an algorithm that takes a curve diagram in D_n and outputs a word in the braid group which takes this curve diagram to the trivial curve diagram. We prove that if a subsurface is an active subsurface for this braid, then as we apply the algorithm:
•The distance between the tangled curve diagram and the trivial curve diagram, projected to the curve graph of the subsurface, remains roughly constant until a certain point where the subsurface becomes active and we start to untangle inside the subsurface.
•Throughout the phase of the algorithm where the subsurface is active, the subsurface is almost round, in a precise sense.
•When the subsurface stops being active, the distance between the tangled curve diagram and the trivial curve diagram is now small and remains small until the algorithm terminates.
We expect this to be of interest both to researchers from the more theoretical geometric group theory community, and to researchers working in combinatorial, algorithmic and computational methods in group theory and topology. It also leads naturally to a computational project, to produce software visualizing the algorithm and illustrating our results. This could be a future project for a student.
•The distance between the tangled curve diagram and the trivial curve diagram, projected to the curve graph of the subsurface, remains roughly constant until a certain point where the subsurface becomes active and we start to untangle inside the subsurface.
•Throughout the phase of the algorithm where the subsurface is active, the subsurface is almost round, in a precise sense.
•When the subsurface stops being active, the distance between the tangled curve diagram and the trivial curve diagram is now small and remains small until the algorithm terminates.
We expect this to be of interest both to researchers from the more theoretical geometric group theory community, and to researchers working in combinatorial, algorithmic and computational methods in group theory and topology. It also leads naturally to a computational project, to produce software visualizing the algorithm and illustrating our results. This could be a future project for a student.