Nonlinear spectral calculus and super-expanders

Abstract

Nonlinear spectral gaps with respect to uniformly convex normed spaces are shown to satisfy a spectral calculus inequality that establishes their decay along Cesaro averages. Nonlinear spectral gaps of graphs are also shown to behave sub-multiplicatively under zigzag products. These results yield a combinatorial construction of super-expanders, i.e., a sequence of 3-regular graphs that does not admit a coarse embedding into any uniformly convex normed space.

Typos fixed based on referee comments. Some of the results of this paper were announced in arXiv:0910.2041. The corresponding parts of arXiv:0910.2041 are subsumed by the current paper

References in corpus (4)

• Euclidean quotients of finite metric spaces
• Expanders with respect to Hadamard spaces and random graphs
• Spectral calculus and Lipschitz extension for barycentric metric spaces
• L_1 embeddings of the Heisenberg group and fast estimation of graph isoperimetry

Cited by in corpus (24)

• Expanders with respect to Hadamard spaces and random graphs
• Spectral calculus and Lipschitz extension for barycentric metric spaces
• Approximation properties for noncommutative $L^p$-spaces of high rank lattices and nonembeddability of expanders
• Relative expanders
• Towards Strong Banach property (T) for SL(3,R)
• Banach space actions and $L^2$-spectral gap
• Superexpanders from group actions on compact manifolds
• From the geometry of box spaces to the geometry and measured couplings of groups
• An average John theorem
• Logarithmic girth expander graphs of $S{L}_n({\mathbb{F}}_p)$
• Nonpositive curvature is not coarsely universal
• Snowflake universality of Wasserstein spaces
• The Baum-Connes conjecture with coefficients for word-hyperbolic groups (after Vincent Lafforgue)
• Dimension independent Bernstein-Markov inequalities in Gauss space
• Vertical versus horizontal Poincaré inequalities on the Heisenberg group
• Rigidity of warped cones and coarse geometry of expanders
• Metric $X_p$ inequalities
• Banach property (T) for $\mathrm{S}{\mathrm{L}}_{\mathrm{n}}(\mathbb{Z})$ and its applications
• From Average Embeddings To Nearest Neighbor Search
• Uniform estimates of nonlinear spectral gaps
• Markov type and threshold embeddings
• A duality operators/Banach spaces
• About the difficulty to prove the Baum Connes conjecture without coefficient for a non-cocompact lattice in $S{p}_4$ in a local field
• Talagrand's influence inequality revisited