Groups, posets, and complexes

The project will study problems at the intersection of topology, algebra, and combinatorics. The main themes of the project are motivated by posets and simplicial complexes arising in group theory. One theme concerns the "universal G-geometry" of the coset poset of a finite group. The PI and his coauthors propose to further their results on the topology of this poset. Related techniques, with a more algebraic geometry flavor, will shed new light on generalized set intersection problems. Another main goal is a minimally classification-dependent proof that the subgroup lattices of nonabelian finite simple groups are not sequentially Cohen-Macaulay. This would make effective a new characterization of solvable groups, highly orthogonal to existing characterizations, and likely would yield better techniques for demarcating between complexes that are sequentially Cohen-Macaulay and those that are not.