Related papers: Groups with non-simply connected asymptotic cones

Gromov asked whether an asymptotic cone of a finitely generated group was always simply connected or had uncountable fundamental group. We prove that Gromov's dichotomy holds for asympotic cones with cut points, as well as, HNN extensions…

Given a bi-invariant metric on a group, we construct a version of an asymptotic cone without using ultrafilters. The new construction, called the directional asymptotic cone, is a contractible topological group equipped with a complete…

We exhibit an infinite family of snowflake groups all of whose asymptotic cones are simply connected. Our groups have neither polynomial growth nor quadratic Dehn function, the two usual sources of this phenomenon. We further show that each…

We construct a finitely presented group with infinitely many non-homeomorphic asymptotic cones. We also show that the existence of cut points in asymptotic cones of finitely presented groups does, in general, depend on the choice of scaling…

We prove that Abels' group over an arbitrary nondiscrete locally compact field has a quadratic Dehn function. As applications, we exhibit connected Lie groups and polycyclic groups whose asymptotic cones have uncountable abelian fundamental…

Using methods from nonstandard analysis, we will discuss which metric spaces can be realized as asymptotic cones. Applying the results we will find in the context of groups, we will prove that a group with "a few" separable asymptotic cones…

We show that whether loops can be shortcut in a group's Cayley graph depends on the choice of finite generating set. Our example is the direct product of two rank-2 free groups and a consequence is that this group has asymptotic cones with…

We give an example of a finitely presented group $G$ with two non-$\pi_1$-equivalent asymptotic cones.

Iterated asymptotic cones have been used by Dru\c{t}u and Sapir to construct a group with uncountably many pairwise non-homeomorphic asymptotic cones. In this paper we define a class of metric spaces which display a wide range of behaviors…

We compute the covering dimension the asymptotic cone of a connected Lie group. For simply connected solvable Lie groups, this is the codimension of the exponential radical. As an application of the proof, we give a characterization of…

We introduce cone bilipschitz equivalences between metric spaces. These are maps, more general than quasi-isometries, that induce a bilipschitz homeomorphism between asymptotic cones. Non-trivial examples appear in the context of Lie…

In the present paper we prove lemmata on strong contractibility in asymptotic cones and metric ultraproducts which we apply to both the case of finitely generated word norms and the case of conjugation invariant norms. We recover…

We construct easy embeddings of relatively free groups (say the free Burnside group, the free solvable group) into finitely presented groups. We introduce a concept of verbal isoperimetric function of a group variety. We prove that if the…

We construct nonlinear hyperbolic groups which are large, torsion-free, one-ended, and admit a finite $K(\pi,1)$. Our examples are built from superrigid cocompact rank one lattices via amalgamated free products and HNN extensions.

Given a principal bundle with a connection, we look for an asymptotic expansion of the holonomy of a loop in terms of its length. This length is defined relative to some Riemannian or sub-Riemannian structure. We are able to give an…

Let G be a connected semisimple Lie group with at least one absolutely simple factor S such that R-rank(S) is at least 2, and let $\Gamma$ be a uniform lattice in G. (a) If $CH$ holds, then $\Gamma$ has a unique asymptotic cone up to…

We introduce the group-compact coarse structure on a Hausdorff topological group in the context of coarse structures on an abstract group which are compatible with the group operations. We develop asymptotic dimension theory for the…

We study the isoperimetric structure of Riemannian manifolds that are asymptotic to cones with non-negative Ricci curvature. Specifically, we generalize to this setting the seminal results of G. Huisken and S.-T. Yau on the existence of a…

We provide new examples of the asymptotic counting for the number of subsets on groups of given size which are free of certain configurations. These examples include sets without solutions to equations in non-abelian groups, and linear…

We describe the (minimal) tree-graded structure of asymptotic cones of non-geometric graph manifold groups, and as a consequence we show that all said asymptotic cones are bilipschitz equivalent. Combining this with geometrization and other…