Topology and Group Theory Seminar: February 5, 2025
Speaker: Itamar Vigdorovich (UCSD)
Title: Effective mixed identity freeness for higher rank lattices and applications to C*-algebras
Abstract: An identity on a group G is a word w that holds throughout the entire group. If w is allowed to include coefficients from G (not just variables), it is called a mixed identity. We show that a lattice Γ in PSL(n, R) has no non-trivial mixed identities in a quantitative and uniform manner: for any r, there exists an element γ of linear length in r that violates all non-trivial mixed words of length at most r. This has powerful C*-algebraic applications due to the recent breakthrough of Amrutam, Gao, Kunnawalkam-Elayavalli, and Patchell. Indeed, when combined with the rapid decay property (which is known, for example, for all cocompact lattices in SL(3, R)), we deduce that the reduced C*-algebra of the lattice satisfies strict comparison, stable rank 1, K0-stability under ultrapowers, uniqueness of the Jiang-Su embedding, and other key properties essential for C*-classification purposes.
Host: Denis Osin