Embeddability of generalized wreath products and box spaces

Given two finitely generated groups that coarsely embed into a Hilbert space, it is known that their wreath product also embeds coarsely into a Hilbert space. We introduce a wreath product construction for general metric spaces X,Y,Z and derive a condition, called the (delta-polynomial) path lifting property, such that coarse embeddability of X,Y and Z implies coarse embeddability of X\wr_Z Y. We also give bounds on the compression of X\wr_Z Y in terms of delta and the compressions of X,Y and Z. Next, we investigate the stability of the property of admitting a box space which coarsely embeds into a Hilbert space under the taking of wreath products. We show that if an infinite finitely generated residually finite group H has a coarsely embeddable box space, then G\wr H has a coarsely embeddable box space if G is finitely generated abelian. This leads, in particular, to new examples of bounded geometry coarsely embeddable metric spaces without property A

Expander graphs and where to find them

Graphs are mathematical objects composed of a collection of “dots” called vertices, some of which are joined by lines called edges. Graphs are ideal for visually representing relations between things, and mathematical properties of graphs can provide an insight into real-life phenomena. One interesting property is how connected a graph is, in the sense of how easy it is to move between the vertices along the edges. The topic dealt with here is the construction of particularly well-connected graphs, and whether or not such graphs can happily exist in worlds similar to ours

Expanders and Property A

We give a cohomological characterisation of expander graphs, and use it to give a direct proof that expander graphs do not have Yu's property A

Uniform Roe algebras of uniformly locally finite metric spaces are rigid

We show that if $X$ and $Y$ are uniformly locally finite metric spaces whose uniform Roe algebras, $\mathrm{C}^*_u(X)$ and $\mathrm{C}^*_u(Y)$, are isomorphic as $\mathrm{C}^*$-algebras, then $X$ and $Y$ are coarsely equivalent metric spaces. Moreover, we show that coarse equivalence between $X$ and $Y$ is equivalent to Morita equivalence between $\mathrm{C}^*_u(X)$ and $\mathrm{C}^*_u(Y)$. As an application, we obtain that if $\Gamma$ and $\Lambda$ are finitely generated groups, then the crossed products $\ell_\infty(\Gamma)\rtimes_r\Gamma$ and $\ell_\infty(\Lambda)\rtimes_r\Lambda$ are isomorphic if and only if $\Gamma$ and $\Lambda$ are bi-Lipshitz equivalent

On the structure of asymptotic expanders

In this paper, we study several analytic and graph-theoretic properties of asymptotic expanders. We show that asymptotic expanders can be characterised in terms of their uniform Roe algebra. Moreover, we use asymptotic expanders to provide uncountably many new counterexamples to the coarse Baum–Connes conjecture. Finally, we show that vertex-transitive asymptotic expanders are actually expanders. In particular, this gives a C ⁎-algebraic characterisation of expanders for vertex-transitive graphs. We achieve these results by showing that a sequence of asymptotic expanders always admits a “uniform exhaustion by expanders”. This also implies that asymptotic expanders cannot be coarsely embedded into any L p-space. </p