Mapping class groups, curve complexes, and Teichmueller spaces

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Mapping class groups and Teichmueller spaces play a large role in our understanding of geometry and topology.Topology is the study of shapes and spaces without their geometry, so one is free to bend and stretch the shapes one is interested in but not tear them. The mapping class groups are the topological symmetries of surfaces. The most vivid examples of mapping class groups are the braid groups. These are the symmetries of a disc with holes.There are several ways of thinking about the braid groups. Imagine the surface of a viscous fluid in a pot in which rods are immersed. One can interchange the rods, without removing them, which stirs the fluid. The rods can return to their starting positions but the surface of the fluid has changed; it has been mixed. The surface of the fluid has undergone a topological symmetry. We may also regard the braid groups as tangles of string in 3-dimensional space. Instead of rods in a fluid, we can start with vertical strings whose lowest points are glued to the base of the pot. By taking hold of the tops of the strings, we can perform the same movements and transpositions as we did with the rods, and this tangles the strings up: it produces braids. These two different perspectives are equivalent. In science, the 3-dimensional point of view of braid groups is applied to polymers, and strands of DNA. The 2-dimensional point of view is applied to topological quantum computing and robotics. The mapping class groups are examples of abstract, algebraic objects called groups. A group is a collection of symmetries: if numbers measure size then groups measure symmetry. Surprisingly, we can learn much about a group by realizing it as the symmetries of a geometric space: this is called geometric group theory. One such useful geometric space for studying the mapping class group is the Teichmueller space.Today, the fast growing area of geometric group theory plays an indispensable role in the recent advances of diverse fields including the geometry and topology of 3-manifolds, complex dynamics, combinatorial group theory, representation theory, logic, and algebraic geometry. Furthermore, there are notions from geometric group theory that are used in large data analysis.The purpose of this project is to implement the far-reaching techniques of geometric group theory to study the mapping class groups and the Teichmueller spaces, which are fundamental objects associated to surfaces. More specifically, we aim to use notions from the latest breakthroughs in 3-manifold theory to study the mapping class groups, and use concepts such as the curve complex---which have provided major developments in hyperbolic geometry---to investigate the Teichmueller space.