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Presented By: Geometry Seminar - Department of Mathematics

Wall structures and curtain models of CAT(0) spaces

Abdul Zalloum

Two of the most classical topics of study in geometric group theory are mapping class groups and CAT(0) cube complexes. This is in part because they both admit powerful combinatorial-like structures encoding interesting aspects of their geometries: curve graphs for the former and hyperplanes for the later. The broad class of CAT(0) spaces -- while also studied extensively in the literature-- generally lacks an intrinsic combinatorial structure similar to that present in cube complexes or mapping class groups. I will talk about recent work with Petyt and Spriano where we introduce two combinatorial objects for studying CAT(0) spaces: curtains which are analogues of cubical hyperplanes and the curtain model which is a counter part of the curve graph. Such structures allow for vast extensions of theorems known in the above contexts to that of CAT(0) spaces including an Ivanov-stlyle rigidity theorem, a dichotomy of a rank-rigidity flavor and the presence of a universal hyperbolic space for rank-one elements.

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