Related papers: Separable and tree-like asymptotic cones of groups

Trees of finite cone type have appeared in various contexts. In particular, they come up as simplified models of regular tessellations of the hyperbolic plane. The spectral theory of the associated Laplacians can thus be seen as induced by…

A certain class of directed metric graphs is considered. Asymptotics for a number of possible endpoints of a random walk at large times is found.

The sizes of subsets of the natural numbers are typically quantified in terms of asymptotic (linear) and logarithmic densities. These concepts have been generalized to weighted $w$-densities, where a specific weight function $w$ plays a key…

We study the conditions under which the isometry of spaces with metrics generated by weights given on the edges of finite trees is equivalent to the isomorphism of these trees. Similar questions are studied for ultrametric spaces generated…

In this article we study asymptotic slopes of strongly semistable vector bundles on a smooth projective surface. A connection between asymptotic slopes and strong restriction theorem of a strongly semistable vector bundle is shown. We also…

The paper is devoted to the large scale geometry of the Heisenberg group $\mathbb H$ equipped with left-invariant Riemannian distances. We prove that two such distances have bounded difference if and only if they are asymptotic, i.e., their…

We prove an analog for integrable measurable cocycles of Pansu's differentiation theorem for Lipschitz maps between Carnot-Carath\'eodory spaces. This yields an alternative, ergodic theoretic proof of Pansu's quasi-isometric rigidity…

We construct cocompact lattices in a product of trees which are not virtually torsion-free. This gives the first examples of hierarchically hyperbolic groups which are not virtually torsion-free

We prove an asymptotic analog of the classical Hurewicz theorem on mappings which lower dimension. This theorem allows us to find sharp upper bound estimates for the asymptotic dimension of groups acting on finite dimensional metric spaces…

In this note we establish the existence of all Curtis-Tits groups and Phan groups with $3$-spherical diagram as classified previously and investigate some of their geometric and group theoretic properties. Whereas it is known that…

The functions $F_{G}(n)$ measures the asymptotic behavior of residual finiteness for a finitely generated group $G$. In previous work \cite{Pengitore_1}, the author claimed a characterization for $F_{N}(n)$ when $N$ is a finitely generated…

Let Gamma be a finitely generated nilpotent group. We consider three closely related problems: (i) the asymptotic cone for an equivariant ergodic family of inner metrics on Gamma, generalizing Pansu's theorem; (ii) the limit shapes for…

We study the class of groups having the property that every non-nilpotent subgroup is equal to its normalizer. These groups are either soluble or perfect. We completely describe the structure of soluble groups and finite perfect groups with…

We consider growing random recursive trees in random environment, in which at each step a new vertex is attached (by an edge of a random length) to an existing tree vertex according to a probability distribution that assigns the tree…

Let X be an arbitrary hyperbolic geodesic metric space and let G be a countable non-elementary weakly acylindrical group of isometries of X. We show that the second bounded cohomology group of G with real coefficients or with coefficients…

We propose a new model for random quotients of groups using independent random walks. In this model, we show that random quotients of acylindrical hyperbolic groups asymptotically almost surely remain acylindrically hyperbolic. Our main…

For every $d\ge 3$, we construct a noncompact smooth $d$-dimensional Riemannian manifold with strictly positive sectional curvature without isoperimetric sets for any volume below $1$. We construct a similar example also for the relative…

Let G be a finitely presented group, and G' its commutator subgroup. Let C be the Cayley graph of G' with all commutators in G as generators. Then C is large scale simply connected. Furthermore, if G is a torsion-free nonelementary…

We study the symmetry properties of the weak positive solutions to a class of quasi-linear elliptic problems having a variational structure. On this basis, the asymptotic behaviour of global solutions of the corresponding parabolic…

We introduce the idea of semigroup-controlled asymptotic dimension. This notion generalizes the asymptotic dimension and the asymptotic Assouad-Nagata dimension in the large scale. There are also semigroup controlled dimensions for the…