Topology and Geometric Group Theory Seminar
Hilbert's Third Problem asks for sufficient conditions that determine when two polyhedra in three-dimensional Euclidean space are scissors congruent. Classically, the attempts to solve this problem (in this and other geometries) lead into group homology and algebraic K-theory, in a somewhat ad-hoc way. In the last decade, Zakharevich has shown that the presence of K-theory here is not ad-hoc, but is integral to the definition of scissors congruence itself. This leads to a natural notion of "higher" scissors congruence groups, or higher algebraic K-theory of scissors congruence.
In this talk I'll describe an exciting, ongoing program to better understand these higher groups, and to compute them in new cases. The main results so far are a trace map to group homology, a Farrell-Jones isomorphism, a Solomon-Tits theorem, and a new description of scissors congruence K-theory as a Thom spectrum. Much of this is joint work with Anna-Marie Bohmann, Teena Gerhardt, Mona Merling, and Inna Zakharevich.