Related papers: Higher index theory for certain expanders and Grom…

The geometry of inverse semigroups is a natural topic of study, motivated both from within semigroup theory and by applications to the theory of non-commutative $C^*$-algebras. We study the relationship between the geometry of an inverse…

We study different geometric properties on infinite graphs, related to the weak-type boundedness of the Hardy-Littlewood maximal averaging operator. In particular, we analyze the connections between the doubling condition, having finite…

We define a way of approximating actions on measure spaces using finite graphs; we then show that in quite general settings these graphs form a family of expanders if and only if the action is expanding in measure. This provides a somewhat…

We introduce a notion of topological property (T) for \'etale groupoids. This simultaneously generalizes Kazhdan's property (T) for groups and geometric property (T) for coarse spaces. One main goal is to use this property (T) to prove the…

We endow the set of persistence diagrams with the strong topology (the topology of countable direct limit of increasing sequence of bounded subsets considered in the bottleneck distance). The topology of the obtained space is described.…

Let $H$ be a complex Hilbert space. Consider the ortho-Grassmann graph $\Gamma^{\perp}_{k}(H)$ whose vertices are $k$-dimensional subspaces of $H$ (projections of rank $k$) and two subspaces are connected by an edge in this graph if they…

We prove that a random group of the graph model associated with a sequence of expanders has fixed-point property for a certain class of CAT(0) spaces. We use Gromov's criterion for fixed-point property in terms of the growth of n-step…

The purpose of this paper is to investigate the global categorical symmetries that arise when gauging finite higher groups in three or more dimensions. The motivation is to provide a common perspective on constructions of non-invertible…

A Thurston map is a branched covering map from $\S^2$ to $\S^2$ with a finite postcritical set. We associate a natural Gromov hyperbolic graph $\G=\G(f,\mathcal C)$ with an expanding Thurston map $f$ and a Jordan curve $\mathcal C$ on…

We develop a theory of higher order structures in compact abelian groups. In the frame of this theory we prove general inverse theorems and regularity lemmas for Gowers's uniformity norms. We put forward an algebraic interpretation of the…

In this paper, we develop a new index theory for manifolds with polyhedral boundary. As an application, we prove Gromov's dihedral extremality conjecture regarding comparisons of scalar curvatures, mean curvatures and dihedral angles…

We investigate the toroidal expanse of an embedded graph G, that is, the size of the largest toroidal grid contained in G as a minor. In the course of this work we introduce a new embedding density parameter, the stretch of an embedded…

We give a characterization of geometric property (T) for a coarse disjoint union of finite graphs with bounded degree using the idea of noncommutative real algebraic geometry. In the proof, we define a $*$-subalgebra $I_u[X]$ of real…

We show that the space of harmonic functions on a finitely generated infinite group G is finite dimensional if, and only if, G has a finite-index subgroup isomorphic to the integers. A key tool is Wilkie and van den Dries's quantitative…

Let $(1\to N_n\to G_n\to Q_n\to 1)_{n\in\mathbb{N}}$ be a sequence of extensions of finite groups. Assume that the coarse disjoint unions of $(N_n)_{n \in \mathbb{N}}$, $(G_n)_{n \in \mathbb{N}}$ and $(Q_n)_{n \in \mathbb{N}}$ have bounded…

The most useful and interesting line bundles over algebraic curves of a very high genus have the ratio \delta of the degree to the genus close to half-integer values, usually \delta \approx 0, \delta \approx 1/2, or \delta \approx 1; the…

We conjecture that finite graphs with positive Cheeger constant admit a spanning subgraph with positive Cheeger constant and girth proportional to the diameter. We prove this conjecture for regular expander graphs with large expansion. Our…

We study the possibility of applying a finite-dimensionality argument in order to address parts of the Baum-Connes conjecture for finitely generated linear groups. This gives an alternative approach to the results of Guentner, Higson, and…

It is known that every finitely unbranched covering $\alpha:\widetilde{S}_{g(\alpha)}\rightarrow S$ of a compact Riemann surface $S$ with genus $g\geq2$ induces an isometric embedding $\Gamma_{\alpha}$ from the Teichm\"uller space $T(S)$ to…

Consider two independent Erd\H{o}s-R\'enyi $G(N,1/2)$ graphs. We show that with probability tending to $1$ as $N\to\infty$, the largest induced isomorphic subgraph has size either $\lfloor x_N-\varepsilon_N\rfloor$ or $\lfloor…