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Dr. Talia Fernos (Mathematics and Statistics) received new funding from National Science Foundation for the project “Boundaries and Nonpositive Curvature.”

This project aims to provide insight into the world of nonpositive curvature by studying the special class of products of hyperbolic spaces (and groups acting on them). These naturally belong to hierarchically hyperbolic spaces (and hence coarse median spaces). One criticism of the work surrounding HHS and coarse median spaces is that it is unclear as to whether they provide new examples or results. The researchers’ hope is that the natural and fundamental class of products of hyperbolic spaces will give way to new insight. The researchers will also continue my work on CAT(0) cube complexes. In fact, it is their expertise in this work that has lead to several insights about the geometry of products hyperbolic spaces.

More specifically, this project will have parts. First, the researchers will study geometric actions on products of hyperbolic spaces and specifically, if they admit a biautomatic structure. Second, in a joint project with Balasubramanya, researchers will relax the restriction on the action from being geometric to being acylindrical. In this context, researchers ask whether several of the known properties of acylindrically hyperbolic groups can continue in higher rank. Third, continuing the project with Balasubramanya, researchers strengthen the acylindrical condition by requiring a type of irreducibility within the product and study a variety of questions, such as the absence of parabolics and the properties that can be inferred, such as a quadratic isoperimetric inequality. Fourth, and last, researchers will continue my extensive work in boundary theory in several ways. In a joint project with Futer and Hagen, researchers study several natural boundaries associated to a CAT(0) cube complex $X$ and show that there are $\Aut(X)$-equivariant homotopy equivalences between them. Then, in a joint project with Hagen, researchers will find a natural boundary associated to a coarse median space that will be cannonically related to the already existing HHS boundary. Researchers hope that the coarse-median boundary and the HHS boundary will also be homotopy equivalent.

 
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