Groups and Operator Algebras Seminar

Speaker:  Ian Charlesworth (Cardiff University)

Title:  Matricial Approximations, Algebraicity, and Strong 1-boundedness

Abstract:  The non-identity group elements in the group algebra of a sofic group admit matricial approximations by permutation matrices with almost-zero diagonal. Elek and Szabó have used this to establish L\"uck's determinant conjecture for sofic groups, and Shlyakhtenko was later able to use this to give a condition for von Neumann algebras of sofic groups being strongly 1-bounded (a microstates-free-entropic condition due to Jung which implies a paucity of matricial approximations for any generating tuple). I will speak on recent joint work with de Santiago, Hayes, Jekel, Kunnawalkam Elayavalli, and Nelson where we adapt these ideas to tracial *-algebras with generating sets well-approximated by matrices with \emph{algebraic} integer entries and approximately constant diagonals, under assumptions about their Galois conjugates. These weaker assumptions still allow us a close-enough analogue of the determinant conjecture to access Shlyakhtenko's results. As a consequence we are able to reduce the question of strong 1-boundedness of certain graph product von Neumann algebras to a question of L^2-Betti numbers.