Related papers: Box spaces, group extensions and coarse embeddings…

We show that a metric space $X$ that, at every point, has a Gromov-Hausdorff tangent with the splitting property (i.e. every geodesic line splits off a factor $\mathbb{R}$), is universally infinitesimally Hilbertian (i.e. $W^{1,2}(X,\mu)$…

Given a finite-dimensional inner product space $V$ and a group $G$ of isometries, we consider the problem of embedding the orbit space $V/G$ into a Hilbert space in a way that preserves the quotient metric as well as possible. This inquiry…

In this paper we describe a class of highly entangled subspaces of a tensor product of finite dimensional Hilbert spaces arising from the representation theory of free orthogonal quantum groups. We determine their largest singular values…

Having developed a description of indefinite extrinsic symmetric spaces by corresponding infinitesimal objects in the preceding paper we now study the classification problem for these algebraic objects. In most cases the transvection group…

Commutative Hilbertian Frobenius algebras are those commutative semi-group objects in the monoidal category of Hilbert spaces, for which the Hilbert adjoint of the multiplication satisfies the Frobenius compatibility relation, that is, this…

For a class of Riemannian manifolds that include products of arbitrary compact manifolds with manifolds of nonpositive sectional curvature on the one hand, or with certain positive-curvature examples such as spheres of dimension at least 3…

For vector/AdS and dS holography we establish the structure of the emergent Hilbert space. This is done through implementation of finite $N$ trace relations on the infinite collective space. For fermionic theories a finite Hilbert space is…

We introduce and study the operation, called dense amalgam, which to any tuple X_1,...,X_k of non-empty compact metric spaces associates some disconnected perfect compact metric space, denoted $\widetilde\sqcup(X_1,...,X_k)$, in which there…

We provide sufficient conditions for two subgroups of a hierarchically hyperbolic group to generate an amalgamated free product over their intersection. The result applies in particular to certain geometric subgroups of mapping class groups…

We consider a version of the notion of F-inverse semigroup (studied in the algebraic theory of inverse semigroups). We point out that an action of such an inverse semigroup on a locally compact space has associated a natural groupoid…

There is a word metric $d$ on countably generated free group $\Gamma$ such that $(\Gamma,d)$ does not admit a coarse uniform embedding into a Hilbert space.

We give a short and elementary proof of the fact that every metric space of finite asymptotic dimension can be embedded into a finite product of trees.

It is widely accepted that the states of any quantum system are vectors in a Hilbert space. Not everyone agrees, however. The recent paper ``The unphysicality of Hilbert spaces'' by Carcassi, Calder\'on and Aidala is a thoughtful dissection…

We prove that bounded conciseness is a closed property in the space of marked groups. As a consequence, we reformulate a conjecture of Fern\'andez-Alcober and Shumyatsky [7] about conciseness in the class of residually finite groups.

Inspired by a classical theorem of topological dimension theory, we prove that every geodesic metric space of asymptotic dimension $n$ containing a bi-infinite geodesic can be coarsely separated by a subset $S$ of asymptotic dimension equal…

From any two median spaces $X,Y$, we construct a new median space $X \circledast Y$, referred to as the diadem product of $X$ and $Y$, and we show that this construction is compatible with wreath products in the following sense: given two…

We provide examples of finitely generated infinite covolume subgroups of $PSL(2,R)^r$ with a "big" limit set, e.g. that contains an open subset of the geometric boundary. They are given by the so called semi-arithmetic Fuchsian groups…

A coarse embedding of a metric space X into a metric space Y is a map f: X-->Y satisfying for every x, y in X: \phi_1(d(x,y)) \leq d(f(x),f(y)) \leq \phi_2(d(x,y)) where \phi_1 and \phi_2 are nondecreasing functions on [0,\infty) with…

Finite topological spaces are in bijective correspondence with preorders on finite sets. We undertake their study using combinatorial tools that have been developed to investigate general discrete structures. A particular emphasis will be…

We provide an analysis of the dynamics of isometries and semicontractions of metric spaces. Certain subsets of the boundary at infinity play a fundamental role and are identified completely for the standard boundaries of CAT(0)-spaces,…