Search for: All records

We provide a characterization of when a coarse equivalence between coarse disjoint unions of expander graphs is close to a bijective coarse equivalence. We use this to show that if the uniform Roe algebras over metric spaces that are coarse unions of expanders graphs are isomorphic, then the metric spaces must be bijectively coarsely equivalent. 

We study which von Neumann algebras can be embedded into uniform Roe algebras and quasi-local algebras associated with a uniformly locally finite metric space X. Under weak assumptions, these C*-algebras contain embedded copies of certain matrix algebras. We aim to show they cannot contain any other von Neumann algebras. One of our main results shows that the only embedded von Neumann algebras are the “obvious” ones. 

By discretizing an argument of Kislyakov, Naor and Schechtman proved that the 1-Wasserstein metric over the planar grid $\{0,1,\dots <\#comment/>,n{\}}^2$has $L_1$-distortion bounded below by a constant multiple of $\sqrt{\log <\#comment/>n}$. We provide a new “dimensionality” interpretation of Kislyakov’s argument, showing that if $\{G_n{\}}_{n=1}^{\mathrm{\infty <\#comment/>}}$is a sequence of graphs whose isoperimetric dimension and Lipschitz-spectral dimension equal a common number $\mathrm{\delta <\#comment/>}\in <\#comment/>[2,\mathrm{\infty <\#comment/>})$, then the 1-Wasserstein metric over $G_n$has $L_1$-distortion bounded below by a constant multiple of $(\log <\#comment/>|G_n|{)}^{\frac{1}{\mathrm{\delta <\#comment/>}}}$. We proceed to compute these dimensions for $\oslash <\#comment/>$-powers of certain graphs. In particular, we get that the sequence of diamond graphs $\{{\mathsf{D}}_n{\}}_{n=1}^{\mathrm{\infty <\#comment/>}}$has isoperimetric dimension and Lipschitz-spectral dimension equal to 2, obtaining as a corollary that the 1-Wasserstein metric over ${\mathsf{D}}_n$has $L_1$-distortion bounded below by a constant multiple of $\sqrt{\log <\#comment/>|{\mathsf{D}}_n|}$. This answers a question of Dilworth, Kutzarova, and Ostrovskii and exhibits only the third sequence of $L_1$-embeddable graphs whose sequence of 1-Wasserstein metrics is not $L_1$-embeddable. 

In this article we start a systematic study of the bi-Lipschitz geometry of lamplighter graphs. We prove that lamplighter graphs over trees bi-Lipschitzly embed into Hamming cubes with distortion at most 6. It follows that lamplighter graphs over countable trees bi-Lipschitzly embed into l1. We study the metric behaviour of the operation of taking the lamplighter graph over the vertex-coalescence of two graphs. Based on this analysis, we provide metric characterisations of superreflexivity in terms of lamplighter graphs over star graphs or rose graphs. Finally, we show that the presence of a clique in a graph implies the presence of a Hamming cube in the lamplighter graph over it. An application is a characterisation, in terms of a sequence of graphs with uniformly bounded degree, of the notion of trivial Bourgain–Milman–Wolfson type for arbitrary metric spaces, similar to Ostrovskii’s characterisation previously obtained in Ostrovskii (C. R. Acad. Bulgare Sci. 64(6), 775–784 (2011)).