Poincaré inequalities, non-linear spectral calculus and superexpanders

Specialeforsvar: Rasmus Kløvgaard Stavenuiter

Titel: Poincaré inequalities, non-linear spectral calculus and superexpanders 

Abstract: We study coarse geometric properties of expander graph sequences in terms of Poincaré inequalities, which form an obstruction for coarsely embedding such sequences into Hilbert space. This leads to a generalization of expanders, namely the notion of superexpander sequences, which do not embed coarsely into uniformly convex Banach space. We prove that Schreier coset graphs of residually finite groups with Kazhdan’s property (T) are expander sequences. We then prove that being an expander with respect to a Banach space is invariant under sphere equivalence of Banach spaces, as well as a generalization of Matoušek’s extrapolation theorem. Moreover, we prove that expander sequences do not coarsely embed into uniformly curved Banach spaces, which are contained in the class of uniformly convex Banach spaces. Finally, we follow Mendel and Naor's work (IHES 2014) to prove that uniformly convex Banach spaces admit a non-linear spectral calculus which will lead to a combinatorial construction of superexpanders through the zig-zag product of graphs

Vejleder: Magdalena Musat
Censor:   Wojciech Szymanski. SDU